Frame of reference in physics – Inertial and Non-inertial – eSaral
A non-accelerating frame of reference is called an inertial frame of reference. If you want to learn more about the frame of reference in physics. Then keep reading.

## Inertial and accelerated frames of reference

### (a) Inertial frames of reference:

A non-accelerating frame of reference is called an inertial frame of reference. A frame of reference moving with a constant velocity is an inertial frame of reference.
1. All the fundamental laws of physics have been formulated in respect of the inertial frame of reference.
2. All the fundamental laws of physics can be expressed as having the same mathematical form in all the inertial frames of reference.
3. The mechanical and optical experiments performed in an inertial frame in any direction will always yield the same results. It is called the isotropic property of the inertial frame of reference.

#### Examples of inertial frames of reference:

1. A frame of reference remaining fixed w.r.t. distance stars is an inertial frame of reference.
2. A spaceship moving in outer space without spinning and with its engine cut-off is also an inertial frame of reference.
3. For practical purposes, a frame of reference fixed to the earth can be considered as an inertial frame. Strictly speaking, such a frame of reference is not an inertial frame of reference, because the motion of the earth around the sun is accelerated motion due to its orbital and rotational motion. However, due to negligibly small effects of rotation and orbital motion, the motion of earth may be assumed to be uniform and hence a frame of reference fixed to it may be regarded as an inertial frame of reference.

### (b) Non-inertial frame of reference:

An accelerating frame of reference is called a non-inertial frame of reference.

Newton’s laws of motion are not directly applicable in such frames, before application

Note:

A rotating frame of reference is a non-inertial frame of reference because it is also an accelerating one due to its centripetal acceleration.

## Pseudo force

The force on a body due to the acceleration of non-inertial frame is called fictitious or apparent or pseudo

force and is given by $\overrightarrow{\mathrm{F}}=-\mathrm{m} \overrightarrow{\mathrm{a}_{0}}$, where $\vec{a}_{0}$ is acceleration of non-inertial frame with respect to an inertial frame and m is mass of the particle or body.

The direction of pseudo force must be opposite to the direction of acceleration of the non-inertial frame.

When we draw the free body diagram of a mass, with respect to an inertial frame of reference we apply only the real forces (forces that are actually acting on the mass).

But when the free body diagram is drawn from a non-inertial frame of reference a pseudo force

(in addition to all real forces) has to be applied to make the equation $\overrightarrow{\mathrm{F}}=\mathrm{m} \overrightarrow{\mathrm{a}}$ to be valid in this frame also.

So, that’s all from this article. I hope you get the idea about the frame of reference in physics. If you have any confusion related to this article then feel free to ask in the comments section down below.

Motion of bodies connected by strings

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Applications of impulse – Impulse, Physics – eSaral
Hey do you want to learn about the applications of impulse? If so, Then you are at the right place

## Impulse

It is not convenient to measure the varying force of impact. Suppose that $\overrightarrow{\mathrm{F}}$ av is the average force acting during impact and $t$ is the small-time for which the impact lasts.

In such situations, it is found that the quantity Fav. It is very easy to measure.

The quantity $\overrightarrow{\mathrm{F}}_{\text {av }} . \mathrm{t}$ is called impulse.

If a variable force acts on a body or particle for a interval of time $t_{1} \rightarrow t_{2}$

Impules is

$\overrightarrow{\mathrm{I}}=\int_{t_{1}}^{t_{2}} \overrightarrow{\mathrm{F}} \mathrm{dt}=\overrightarrow{\mathrm{p}_{2}}-\overrightarrow{\mathrm{p}_{1}}$

Impules momentum theorem

The integral $\int_{0}^{t} \overrightarrow{\mathrm{F}} \mathrm{dt}$ is measure of the impulse, when the force of impact acts on the body and from equation, we find that it is equal to total change in momentum of the body.

Dimensional formula $\&$ unit. The dimensional formula of impulse is the same as that of momentum i.e. [ML $T^{-1}$ ]. In SI, the unit of impulse is N-s or kg ms $^{-1}$ and in cgs system it is dyne-s or $\mathrm{g} \mathrm{cm} \mathrm{s}^{-1}$

## Applications of Concept of Impulse :

(a) Bogies of a train are provided with buffers: The buffers increase the time interval of jerks during shunting and hence reduce the force with which the bogies pull each other. $(\because \mathrm{F} \Delta \mathrm{t}=\mathrm{constant})$

(b) A cricket player draws his hands back while catching a ball: The drawing back of hands increases time interval and hence reduces force with which the ball hits the hand.

(c) A person jumping on a hard cement floor receives more injuries than a person jumping on a muddy or sandy road: Because on the hard cemented floor the feet of the man immediately comes to rest and the time is small, therefore, the force experienced by the man is large while on the sandy soil the feet embedes into the soil, thus the time taken for same change in momentum is comparatively increased. Thus the force is decreased.

(d) Cars, buses, trucks, bogies of the train, etc are provided with a spring system to avoid severe jerks. Due to the spring system, the time interval of the jerks increases. As the rate of change of momentum will be smaller, comparatively lesser force acts on the passengers during the jerks.

(e) China wares are wrapped in straw or paper before packing. China wares are wrapped in straw or paper so that when they receive jerks during transportation, the time of impact may be more. As such, only a small force can act on the china wares during jerks and there is no fear of damage to them.

(f) If a graph is plotted between $\overrightarrow{\mathrm{F}}$ and $\mathrm{t}$ with $\overrightarrow{\mathrm{F}}$ as ordinate (Y-axis)

and $t$ as $(X$ -axis)

Impulse = Area enclosed by F-t curve and time axis for specified duration.

For example impulse between time $t_{1}$ and $t_{2}=$ area abdc.

Newton’s Laws of Motion

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Ex. A hammer weighing $2.5 \mathrm{kg}$ moving with a speed of $1 \mathrm{m} / \mathrm{s}$ strikes the head of a nail driving it $10 \mathrm{cm}$ into the wall. What is the acceleration during impact and the impulse imparted to wall.

Sol. The initial velocity of nail is same as that of hammer i.e. $\mathrm{u}=1 \mathrm{m} / \mathrm{s}, \mathrm{v}=0, \mathrm{s}=10 \mathrm{cm}=0.10 \mathrm{m}$

$$\begin{array}{l}\therefore \quad v^{2}=u^{2}+2 a s \text { gives } \\0=1^{2}+2 a \times 0.1\end{array}$$

This gives $a=-\frac{1}{0.2}=-5 \mathrm{m} / \mathrm{s}^{2}$

i.e. Retardation $\mathrm{a}=5 \mathrm{m} / \mathrm{s}^{2}$

Impulse imparted to wall = – change in momentum of hammer

$$=-\{\mathrm{M}(\mathrm{v}-\mathrm{u})\}=-\{0.5(0-1)\}=+2.5 \mathrm{N}-\mathrm{s}$$

Ex. Magnitude of the force in newton acting on a body varies with time $t$ (in mili-second) as shown in fig. below. Calculate magnitude of total impulse of the force.

Sol. The impulse

\begin{aligned} \mathrm{I}=\int \mathrm{F} \mathrm{dt} &=\frac{1}{2}(0.014 \mathrm{s}-0.003 \mathrm{s})(10 \mathrm{N}) \\ &=0.055 \mathrm{N} \cdot \mathrm{s}=0.055 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} \end{aligned}

Area of the graph gives the magnitude of the Impulse.

Ex. $\quad$ A force exert an impulse I on a body changing its speed from u to v. The force and object’s motion are along the same line. Show that the work done by the force is $\frac{\mathrm{I}(\mathrm{u}+\mathrm{v})}{2}$.

Sol. According to work-energy theorem we have

$$W=\Delta K=\frac{1}{2} m v^{2}-\frac{1}{2} m u^{2} \quad \text { or } \quad W=\frac{1}{2} m(v-u)(v+u)$$

But according to impulse-momentum theorem

$$\mathrm{I}=\mathrm{m}(\mathrm{v}-\mathrm{u})$$

So, eliminating $\mathrm{m}$ between $\mathrm{eq}^{\mathrm{n}} .(1)$ and (2)

$$W=\frac{1}{2} \frac{I}{(v-u)}(v-u)(v+u)=\frac{I(v+u)}{2} \quad \text { This is the required result. }$$

Ex. A cricket ball of mass $150 \mathrm{g}$ is moving with a velocity of $12 \mathrm{m} / \mathrm{s}$ and is hit by a bat so that the ball is turned back with a velocity of $20 \mathrm{m} / \mathrm{s}$. If the duration of contact between the ball and bat is $0.01 \mathrm{s},$ find ther impulse and the average force exerted on the ball by the bat.

Sol. According to given problem change in momentum of the ball

$$\Delta p=p_{f}-p_{i}=m(v-u)=150 \times 10^{-3}[20-(-12)]$$

So by impulse-momentum theorem $I=\Delta p=4.8 \mathrm{N}-\mathrm{s}$

And by time averaged definition of force in case of impulse $\mathrm{F}_{\mathrm{av}}=\frac{\mathrm{I}}{\Delta \mathrm{t}}=\frac{\Delta \mathrm{p}}{\Delta \mathrm{t}}=\frac{4.80}{0.01}=480 \mathrm{N}$
Horizontal Projectile motion – Motion in Plane – eSaral
Hey, do you want to learn about Horizontal projectile motion? If yes. Then you are at the right place.

## Horizontal Projectile motion

Suppose a body is thrown horizontally from point O, with velocity u. Height of O from ground = H. Let X-axis be along horizontal and Y-axis be vertically downwards and origin O is at point of projection as shown in fig. Let the particle be at P at a time t. The co-ordinates of P are (x, y) Distance travelled along X-axis at time t with uniform velocity i.e. Velocity of projection and without acceleration.

The horizontal component of velocity $\mathrm{V}_{\mathrm{x}}$ = u

and horizontal displacement x = u . t….(1)

to calculate y, consider vertical motion of the projectile

initial velocity in vertical direction $\mathrm{u}_{\mathrm{y}}$ = 0.

acceleration along y direction $a_{y}$ = g (acc. due to gravity)

so

$v_{v}=a_{y} t$

(y comp. of velocity at time t)

Or

$v_{y}=g t$….(2)

(as body were dropped from a height)

Resultant velocity at time t is

$\overrightarrow{\mathrm{v}}=\mathrm{u} \hat{\mathrm{i}}+(\mathrm{gt}) \hat{\mathrm{j}}$

$v=\sqrt{u^{2}+(g t)^{2}}$

if $\beta$ is the angle of velocity with X-axis (horizontal) $\tan \beta=\frac{\mathrm{gt}}{\mathrm{u}}$

and

$y=\frac{1}{2} g t^{2}$……(3)

Or

$y=\frac{1}{2} g\left(\frac{x}{u}\right)^{2}$

[from equation (i) $\left.\mathrm{t}=\frac{\mathrm{x}}{\mathrm{u}}\right]$

Or

$y=\frac{g}{2 u^{2}} \cdot x^{2}$

Or

$y=k x^{2}$

here $\mathrm{k}=\frac{\mathrm{g}}{2 \mathrm{u}^{2}} \quad(\mathrm{k}$ is constant)

This is eqn. of a parabola.

A body thrown horizontally from a certain height above the ground follows a parabolic trajectory till it hits the ground.
1. Time of flight$\mathrm{T}=\sqrt{\frac{2 \mathrm{H}}{\mathrm{g}}}$ [as $\left.\mathrm{y}=\frac{1}{2} \mathrm{gt}^{2}, \mathrm{~T}=\sqrt{\frac{2 \mathrm{H}}{\mathrm{g}}}\right]$

2. Range horizontal distance covered = R.R = u × time of flight$\mathrm{R}=\mathrm{u} \cdot \sqrt{\frac{2 \mathrm{H}}{\mathrm{g}}}$

$\left[\because \mathrm{H}=\frac{\mathrm{g}}{2 \mathrm{u}^{2}} \mathrm{R}^{2}\right]$

3. Velocity when it hits the ground$\mathrm{v}_{\mathrm{g}}=\sqrt{\mathrm{u}^{2}+2 \mathrm{gH}}$

Oblique Projectile motion

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Oblique Projectile motion – Motion in Plane – eSaral
Hey, do you want to learn about the Oblique Projectile motion? If yes. Then keep reading.

## Oblique Projectile motion

Consider the motion of a bullet which is fired from a gun so that its initial velocity $\overrightarrow{\mathrm{u}}$ makes an angle $\theta$ with the horizontal direction. Let us take $X$ -axis along ground and Y-axis along vertical.

$\overrightarrow{\mathrm{u}}$ can be resolved as

$\mathrm{u}_{\mathrm{x}}=\mathrm{u} \cos \theta \quad$ (along horizontal)

$\& u_{y}=u \sin \theta$ (along vertical)

motion of bullet can be resolved into horizontal and vertical motion.

(i) In horizontal direction there is no acc. so it moves with constant velocity $v_{x}=u_{x}=u \cos \theta$

So distance traversed in time $t$ is $x=u_{x} t$ or  $t=\frac{x}{u \cos \theta}$

The motion in the vertical direction is the same as that of a ball thrown upward with an initial velocity $\mathrm{u}_{y}=\mathrm{u} \sin \theta$ and $\mathrm{acc}=-\mathrm{g}$ (downward).

So at time $t$ vertical component of velocity $v_{y}=u_{y}-g t=u \sin \theta-g t$

Displacement along y direction

$y=(u \sin \theta) t-\frac{1}{2} g t^{2}$

Substituting the value of $t$ from eqn. (i) in eqn. (iii)

we get

$\quad y=(u \sin \theta)\left(\frac{x}{u \cos \theta}\right)-\frac{1}{2} g\left(\frac{x}{u \cos \theta}\right)^{2}$

or $\left.y=x \tan \theta-\frac{g}{2 u^{2} \cos ^{2} \theta} \cdot x^{2}\right]$ This is eqn. of parabola.

The trajectory of projectile is parabolic

The projectile will rise to maximum height $\mathrm{H}$ (where $\mathrm{v}_{x}=\mathrm{u} \cos \theta, \mathrm{v}_{y}=0$ ) and then move down again to reach the ground at a distance $\mathrm{R}$ from origin.

Setting $x=R$ and $y=0$ (since projectile reaches ground again)

$\mathrm{O}=\mathrm{R} \tan \theta-\frac{\mathrm{g}}{2 \mathrm{u}^{2} \cos ^{2} \theta} \cdot \mathrm{R}^{2}$

We get $R=\frac{2 u^{2} \cos ^{2} \theta}{g} \times \frac{\sin \theta}{\cos \theta}$

or $\mathrm{R}=\frac{2 \mathrm{u}^{2}}{\mathrm{~g}} \cdot \sin \theta \cos \theta$

or Range $R=\frac{u^{2} \sin 2 \theta}{g}$

If time for upward journey is t

at highest point $\quad v_{y}=0$ so $0=(u \sin \theta)-g t$

$\left(v_{y}=u_{y}-g t\right)$

or $\mathrm{t}=\frac{\mathrm{u} \sin \theta}{\mathrm{g}}$

$\mathrm{T}=2 \mathrm{t} \quad$ (it will take same time for downward journey)

$\therefore$

$T=\frac{2 u \sin \theta}{g}$ Time of flight

At the highest point $y=H$ and $v_{y}=0$ Maximum Height and Time of

Flight Depends on Vertical

Component of Initial Velocity So that $H=\frac{u_{y}^{2}}{2 g} \quad\left[v_{y}^{2}=u_{y}^{2}-2 g y\right]$

or $\mathrm{H}=\frac{\mathrm{u}^{2} \sin ^{2} \theta}{2 \mathrm{~g}}$ Maximum Height

we can also determine $R$ as follows

so $x=u_{x} t$

$R=u_{x} \cdot T$

$=(u \cos \theta)\left(\frac{2 u \sin \theta}{g}\right)$

or $\mathrm{R}=\frac{\mathrm{u}^{2} \sin 2 \theta}{\mathrm{g}}$

velocity at time t $\vec{v}_{t}=v_{x t} \hat{i}+v_{y t} \hat{j}$

$=(u \cos \theta) \hat{i}+(u \sin \theta-g t) \hat{j}$

$v=\sqrt{u^{2} \cos ^{2} \theta+(u \sin \theta-g t)^{2}}$

Note :

(i) Alternative $e q^{n}$, of trajectory $y=x \tan \theta\left(1-\frac{x}{R}\right)$ where $R=\frac{2 u^{2} \sin \theta \cos \theta}{g}$

(ii) Vertical component of velocity $v_{y}=0,$ when particle is at the highest point of trajectory.

(iii) Linear momentum at highest point = mu cos $\theta$ is in horizontal direction.

(iv) Vertical component of velocity is +ive when particle is moving up.

(v) Vertical component of velocity is -ive when particle is moving down.

(vi) Resultant velocity of particle at time $t \vee=\sqrt{v_{x}^{2}+v_{y}^{2}}$ at an angle $\phi=\tan ^{-1}\left(\frac{v_{y}}{v_{x}}\right) .$

(vii) Displacement from origin, $s=\sqrt{x^{2}+y^{2}}$

Special Points :

(1) The three basic equation of motion, i.e.

$v=u+a t$

$\mathrm{s}=\mathrm{ut}+\frac{1}{2} \mathrm{at}^{2} \quad \mathrm{v}^{2}=\mathrm{u}^{2}+2 \mathrm{as}$

For projectile motion give:

$T=\frac{2 u \sin \theta}{g} \quad R=\frac{u^{2} \sin 2 \theta}{g}$

$H=\frac{u^{2} \sin ^{2} \theta}{2 g}$

(2) In case of projectile motion,

The horizontal component of velocity (u $\cos \theta$ ), acceleration (g), and mechanical energy remain constant.

Speed, velocity, vertical component of velocity (u $\sin \theta)$, momentum, kinetic energy and potential energy all change. Velocity and K.E. are maximum at the point of projection, while minimum (but not zero) at the highest point.

(3) If angle of projection is changed from

then range

$\theta-\theta=(90-\theta)$

$R=\frac{u^{2} \sin 2 \theta^{\prime}}{g}=\frac{u^{2} \sin 2(90-\theta)}{g}=\frac{u^{2} \sin 2 \theta}{g}=R$

So a projectile has same range for angles of projection $\theta$ and $(90-\theta)$

But has different time of flight (T), maximum height (H) \& trajectories

Range is also same for $\theta_{1}=45^{\circ}-\alpha$ and $\quad \theta_{2}=45^{\circ}+\alpha .\left[\right.$ equal $\left.\frac{u^{2} \cos 2 \alpha}{g}\right]$

(4) For maximum Range $\quad R=R_{\max } \Rightarrow 2 \theta=90^{\circ}$

$\text { for } \quad \theta=45^{\circ}$

$R_{\max }=\frac{u^{2}}{g} \quad\left[\right.$ For $\sin 2 \theta=1=\sin 90^{\circ}$ or $\left.\theta=45^{\circ}\right]$

When range is maximum $\Rightarrow$ Then maximum height reached

$\mathrm{H}=\frac{\mathrm{u}^{2} \sin ^{2} 45}{2 \mathrm{~g}}\left(\mathrm{When} \mathrm{R}_{\max }\right)$

or

$\mathrm{H}=\frac{\mathrm{u}^{2}}{4 \mathrm{~g}}$

hence maximum height reached (for $\left.R_{\max }\right) \quad H=\frac{R_{\max }}{4}$

(5) For height H to be maximum

$\mathrm{H}=\frac{\mathrm{u}^{2} \sin ^{2} \theta}{2 \mathrm{~g}}=\max \quad \text { i.e. } \sin ^{2} \theta=1(\max ) \text { or for } \theta=90^{\circ}$

So that $\mathrm{H}_{\max }=\frac{\mathrm{u}^{2}}{2 \mathrm{~g}} \quad$ When projected vertically (i.e. at $\left.\theta=90^{\circ}\right)$

in this case Range $R=\frac{u^{2} \sin \left(2 \times 90^{\circ}\right)}{9}=\frac{u^{2} \sin 180^{\circ}}{9}=0$

$\mathrm{H}_{\max }=\frac{\mathrm{u}^{2}}{2 \mathrm{~g}}$ (For vertical projection) and $\mathrm{R}_{\max }=\frac{\mathrm{u}^{2}}{\mathrm{~g}}$ (For oblique projection with same velocity)

so $\mathrm{H}_{\max }=\frac{R_{\max }}{2}$

If a person can throw a projectile to a maximum distance (with $\theta=45^{\circ}$ ) $R_{\max }=\frac{\mathbf{u}^{2}}{\mathrm{~g}}$.
.
The maximum height to which he can throw the projectile (with $\theta=90^{\circ}$ ) $\mathrm{H}_{\max }=\frac{R_{\text {max }}}{2}$

(6) At highest point

Potential energy will be max and equal to $(\mathrm{PE})_{4}=\mathrm{mgH}=\mathrm{mg} \cdot \frac{\mathrm{U}^{2} \sin ^{2} \theta}{2 \mathrm{~g}}$ or $(\mathrm{PE})_{\mathrm{H}}=\frac{1}{2} \mathrm{mu}^{2} \sin ^{2} \theta$

While K.E. will be minimum (but not zero) and at the highest point as the vertical component of velocity is zero.

$(\mathrm{KE})_{\mathrm{H}}=\frac{1}{2} \mathrm{mv}_{\mathrm{H}}^{2}=\frac{1}{2} \mathrm{~m}(\mathrm{u} \cos \theta)^{2}$

$=\frac{1}{2} \mathrm{mu}^{2} \cos ^{2} \theta$

so $(\mathrm{PE})_{\mathrm{H}}+(\mathrm{KE})_{\mathrm{H}}$

$=\frac{1}{2} m u^{2} \sin ^{2} \theta+\frac{1}{2} m u^{2} \cos ^{2} \theta$

= $\frac{1}{2} \mathrm{mu}^{2}=$ Total M.E.

So in projectile motion mechanical energy is conserved.

$\left(\frac{P E}{K E}\right)_{H}=\frac{\frac{1}{2} m u^{2} \sin ^{2} \theta}{\frac{1}{2} m u^{2} \cos ^{2} \theta}=\tan ^{2} \theta$

(7) In case of projectile motion if range $R$ is $n$ times the maximum height $H,$ l.e. $R=n H$

then $\frac{u^{2} \sin 2 \theta}{g}=n \cdot \frac{u^{2} \sin ^{2} \theta}{2 g}$

or $2 \cos \theta=\frac{n \sin \theta}{2}$

or $\tan \theta=\frac{4}{n} \quad \Rightarrow \quad \theta=\tan ^{-1}\left(\frac{4}{n}\right)$

(8) Weight of a body in projectile motion is zero as it is a freely falling body.

So, that’s all from this article. I hope you get the idea about the Oblique projectile motion. If you liked this explanation then please share it with your friends.

Motion in Two Dimension

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Motion in two dimensions – Motion in Plane – eSaral

## Motion in Two Dimensions

An object moving in a plane is said to have two-dimensional motion. The two-dimensional motion is equal to the vector sum of two one-dimensional motions along a mutually perpendicular direction.

Let the position of point P at a time t be given by position

vector $\overrightarrow{\mathrm{OP}}=\overrightarrow{\mathrm{r}}$

$\vec{r}=\hat{i} r \cos \theta+\hat{j} r \sin \theta$

$=\hat{i} x+\hat{j} y$

## Displacement

Let the position of point P at time $\mathrm{t}_{1}$ be described by position vector $\vec{r}_{1}=x_{1} \hat{i}+y_{1} \hat{j}$ and at time $\mathrm{t}_{2}$

position Q is given by position vector $\vec{r}_{2}=x_{2} \hat{i}+y_{2} \hat{j}$

from $\Delta \mathrm{OPQ} \Rightarrow \overrightarrow{\mathrm{OP}}+\overrightarrow{\mathrm{PQ}}=\overrightarrow{\mathrm{OQ}}$

or $\quad \overrightarrow{\mathrm{PQ}}=\overrightarrow{\mathrm{OQ}}-\overrightarrow{\mathrm{OP}}$

Displacement $\overrightarrow{P Q}=\delta \vec{r}=\vec{r}_{2}-\vec{r}_{1}$

in time interval $\delta \mathrm{t}=\left(\mathrm{t}_{2}-\mathrm{t}_{1}\right)$

or

$\vec{\delta}_{r}=\left(x_{2} \hat{i}+y_{2} \hat{j}\right)-\left(x_{1} \hat{i}+y_{1} \hat{j}\right)$

$=\left(x_{2}-x_{1}\right) \hat{i}+\left(y_{2}-y_{1}\right) \hat{j}$

$=\delta \mathrm{x} \hat{\mathrm{i}}+8 \mathrm{y} \hat{\mathrm{j}}$

displacement along X-axis $\delta x=x_{2}-x_{1}$

displacement along Y-axis $\delta y=y_{2}-y_{1}$

Thus, displacement in 2 dimensions is equal to the vector sum of two one dimensional displacements along mutually perpendicular directions.

Let particle move with uniform velocity at $\overrightarrow{\mathrm{V}}$ an angle $\theta$ with X-axis.

Then in component form $\overrightarrow{\mathrm{v}}=\mathrm{v}_{\mathrm{x}} \hat{\mathrm{i}}+\mathrm{v}_{\mathrm{y}} \hat{\mathrm{j}}$

here $v_{x}=v \cos \theta$

and $\quad v_{y}=v \sin \theta$

and $\quad \delta \mathrm{x}=\mathrm{v}_{\mathrm{x}} \delta \mathrm{t}$

$\delta \mathrm{y}=\mathrm{v}_{\mathrm{y}} \delta \mathrm{t}$

or $\delta x=(v \cos \theta) \delta t$

$\delta y=(v \sin \theta) \delta t$

so with $\mathrm{V}_{\mathrm{x}}$ we get displacement along X-axis only and $\mathrm{v}_{\mathrm{y}}$ gives displacement along Y-axis only.

And if particle is moving with uniform acceleration $\overrightarrow{\mathrm{a}}$, then

$\overrightarrow{\mathrm{a}}=\mathrm{a}_{\mathrm{x}} \hat{\mathrm{i}}+\mathrm{a}_{\mathrm{y}} \hat{\mathrm{j}}$

If direction of $\vec{a}$ makes angle $\phi$ with X-axis then $a_{x}=a \cos \phi$ and $\mathrm{a}_{\mathrm{y}}=\mathrm{a} \sin \phi$

are components of $\overrightarrow{\mathrm{a}}$.

Due to $\mathrm{a}_{\mathrm{x}}$, there is a change in the X component of velocity only with no change in Y-component.

Similarly, $\mathrm{a}_{\mathrm{y}}$ will change only the Y component of velocity at time t

So $v_{x}=u_{x}+a_{x} t$

(here $\mathrm{u}_{\mathrm{x}}$ and $\mathrm{u}_{\mathrm{y}}$ are components of initial velocity)

And

$v_{y}=u_{y}+a_{y} t$

Hence

$v_{x} \hat{i}+v_{y} \hat{j}=\left(u_{x}+a_{x} t\right) \hat{i}+\left(u_{y}+a_{y} t\right) \hat{j}$

$=\left(u_{x} \hat{i}+u_{y} \hat{j}\right)+\left(a_{x} \hat{i}+a_{y} \hat{j}\right) t$

Or

$\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{u}}+\overrightarrow{\mathrm{a}} \mathrm{t}$

and similarly, component of displacement are

$s_{x}=u_{x} t+\frac{1}{2} a_{x} t^{2}$

And

$s_{y}=u_{y} t+\frac{1}{2} a_{y} t^{2}$

Hence

$s_{x} \hat{i}+s_{y} \hat{j}=\left(u_{x} \hat{i}+u_{y} \hat{j}\right)+\frac{1}{2}\left(a_{x} \hat{i}+a_{y} \hat{j}\right) t^{2}$

Or

$\vec{s}=\vec{u} t+\frac{1}{2} \vec{a} t^{2}$

Newtons Laws of Motion

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Hey, students do you want to know how to state the law of radioactive decay? If yes. Then read this article till the end.

The rate of decay (number of disintegrations per second) is proportional to number of radioactive atoms (N) present at that time t

rate of decay $\frac{-\mathrm{d} \mathrm{N}}{\mathrm{dt}} \propto \mathrm{N}$

or $\frac{\mathrm{d} \mathrm{N}}{\mathrm{dt}}=-\lambda \mathrm{N}$

or $\quad N=N_{0} e^{-\lambda t}$…..(1)

where $\lambda$ is disintegration constant, $\mathrm{N}_{0}$ = number of active atoms at t = 0
1. Equation one is the radioactive decay law. It shows that the number of active nuclei decreases exponentially with time.
2. The fraction of active atoms remaining at time t is$\frac{\mathrm{N}}{\mathrm{N}_{0}}=\mathrm{e}^{-\lambda \mathrm{t}}$
3. The number of atoms that have decayed in time t is$N_{0}-N$

$=N_{0}\left(1-e^{-\lambda t}\right)$
4. The fraction of atoms that have decayed in time t is$\frac{N_{0}-N}{N_{0}}$

$=1-e^{-\lambda t}$

## Decay constant

1. Decay constant$\lambda=\frac{-\mathrm{d} \mathrm{N} / \mathrm{dt}}{\mathrm{N}}$

$=\frac{\text { rate of decay }}{\text { number of active atoms }}$

2. at $t=\frac{1}{\lambda}$$\mathrm{N}=\frac{\mathrm{N}_{0}}{\mathrm{e}} The decay constant of a radioactive element is equal to the reciprocal of the time after which the number of remaining active atoms reduces to \frac{1}{\mathrm{e}} times of original value. 3. at t=\frac{1}{\lambda}fraction of active nuclei left \frac{\mathrm{N}}{\mathrm{N}_{0}}=\frac{1}{\mathrm{e}}=0.37 or 37 \% fraction of decayed nuclei 1-\frac{N}{N_{0}} =0.63=63 \% 4. \lambda=\frac{\mathrm{dN} / \mathrm{N}}{\mathrm{dt}} The decay constant is the probability of decay per active atom per unit time. 5. The decay constant depends on the nature of the radioactive substance and is independent of temperature, pressure, force, etc. 6. The decay constant for a stable substance is zero 7. Unit of decay constant is second ^{-1} and dimension is \mathrm{T}^{-1} 8. If there are more than one radioactive elements in a group then the resultant decay constant is equal to sum of individual decay constants \lambda=\lambda_{1}+\lambda_{2}+\lambda_{3}+…….. or \frac{1}{\mathrm{~T}}=\frac{1}{\mathrm{~T}_{1}}+\frac{1}{\mathrm{~T}_{2}}+\ldots ## Half life The time in which the number of radioactive atoms reduce to half of its initial value is known as half-life i.e. at \mathrm{t}=\mathrm{T} N=\frac{N_{0}}{2} from radioactive decay law \frac{N_{0}}{2}=N_{0} e^{-\lambda T} or \mathrm{T}=\frac{0.693}{\lambda} 1. The half-life depends on the nature of radioactive elements. 2. The half-life of an element indicates the rate of decay. When half-life is a large rate of decay is small. 3. After t = nT number of active atoms leftN=\frac{N_{0}}{2^{n}} =\frac{1}{2^{t / T}} \cdot \mathrm{N}_{0} where T = half-life and n = number of half-lives. 4. Number of radioactive atoms decayed in n half-livesN_{0}-\frac{N_{0}}{2^{n}} =N_{0}\left(\frac{2^{n}-1}{2^{n}}\right) 5. The half-life for a given radioactive substance is constant. It does not change with time. It is unaffected by pressure, temperature, etc. so, that’s all from this blog, I hope you get the idea about how to state the law of radioactive decay. If you found this article informative then please share it with your friends. Also, read Uses Of radioactive Isotopes To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. Uses of radioactive isotopes – Radioactivity, Physics – eSaral Hey, do you want to learn about the uses of radioactive isotopes? If yes. Then you are at the right place. ## Uses of radioactive isotopes 1. In Medicine a. \mathrm{Co}^{60} for treatment of cancerb. \mathrm{Na}^{24} for circulation of blood c. \mathbf{I}^{131} for thyroid d. S r^{90} for treatment of skin & eye e. \mathrm{Fe}^{59} for location of brain tumor f. Radiographs of castings and teeth 2. In Industries a. For detecting leakage in water and oil pipe lines b. For investigation of wear & tear, study of plastics & alloys, thickness measurement. 3. In Agriculture a. \mathrm{C}^{14} to study kinetics of plant photosynthesis b. \mathrm{P}^{32} to find nature of phosphate which is best for given soil & crop c. \mathrm{Co}^{60} for protecting potato crop from earth worm d. sterilization of insects for pest control. 4. radioactive isotopes a. \mathrm{K}^{40} to find age of meteorites b. S^{35} in factories 5. Carbon dating a. It is used to find age of earth and fossils b. The age of earth is found by Uranium disintegration and fossil age by disintegration of \mathrm{C}^{14}. c. The estimated age of earth is about 5 \times 10^{9} years. d. The half-life of \mathrm{C}^{14} is 5700 years. 6. As Tracers a. A very small quantity of radio isotope present in any specimen is called tracer. b. This technique is used to study complex biochemical reactions, in detection of cracks, blockages etc, tracing sewage or silt in sea. 7. In Geology a. For dating geological specimens like ancient rocks, lunar rocks using Uranium. b. For dating archaeological specimens, biological specimens using \mathrm{C}^{14}. So, that’s it from this article. I hope you get the idea about the uses of radioactive isotopes. If you liked this explanation then please share it with your friends. Also, see Nuclear Fission and Fusion To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. Working of NPN and PNP transistor – Electronics, Physics – eSaral Hey, do you want to learn about the Working of NPN and PNP transistor? if yes. Then you are at the right place. ## P-N-P Transistor: The holes of the P region (emitter) are rippled by the positive terminal of battery \mathrm{V}_{\mathrm{EE}} towards the base. The potential barrier at the emitter junction is reduced as it is forward bias and hence the holes cross this junction and penetrate into the N region. This constitutes the emitter current. The width of the base region is very thin and it is lightly doped and hence only two to five percent of the holes recombine with the free electrons of the N region. This constitutes the base current \mathrm{I}_{\mathrm{B}}. which of course, is very small. The remaining holes (95% to 98%) are able to drift across the base and enter into the collector region. This constitutes the collector current. ## N-P-N Transistor: Working: The electron in the emitter region is rippled from the negative terminal of the battery towards the emitter junction. Since the potential barrier at the junction is reduced due to forward bias and the base region is very thin and lightly doped, electrons cross the p-type base region. A few electrons combine with the holes in P-region and are lost as the charge carriers. Now the electrons in N-region (collector region) readily swept up by the positive collector voltage \mathrm{V}_{\mathrm{CC}}. The current conduction in the N-P-N transistor is carried out by electrons. ## Transistor configuration Three types of transistor circuit configuration are Common base (CB) Common emitter (CE) Common collector (CC) The term ‘Common’ is used to denote the transistor lead which is common to the input and the output. ### Common Base Configuration In this configuration the input signal is applied between emitter and base and the output is taken from collector and base. In common base as an amplifier phase difference between input and output is zero. ## Characteristics: ### Input characteristic: The curve between emitter current \mathrm{I}_{\mathrm{E}} and emitter base voltage V_{\mathrm{EB}} at constant collector base voltage \mathrm{V}_{\mathrm{CB}} represents the input characteristics. Collector base voltage \mathrm{V}_{\mathrm{CB}} is kept fixed. • There exist a cut in, offset or threshold voltage V_{\mathrm{EB}} below which the emitter current is very small. • The emitter current \mathrm{I}_{\mathrm{E}} increases rapidly with small increase in emitter-base voltage V_{\mathrm{EB}}. This shows that the input resistance is very small. ### Output characteristic: Curve between collector current \mathrm{I}_{\mathrm{C}} and collector base voltage V_{\mathrm{CB}} at constant emitter \mathrm{I}_{\mathrm{E}} represents the output characteristic. ## Common emitter configuration In this configuration the input signal is applied between base and emitter and the output is taken from collector and emitter. In common emitter as an amplifier phase difference between input and output is \pi. ## Characteristic ### Input characteristics: The curve between base current \mathrm{I}_{\mathrm{B}} and base emitter voltage \mathrm{V}_{\mathrm{BE}} at constant collector-emitter voltage \mathrm{V}_{\mathrm{CE}} represents the input characteristic. In this case, \mathrm{I}_{\mathrm{B}} increases less rapidly with V_{B E} as compared to common-base configuration. This shows input resistance of the common-emitter circuit is higher than that of the common-base circuit. ### Output characteristic: The curve between collector current \mathrm{I}_{\mathrm{C}} and collector emitter voltage V_{C E} at constant base current \mathrm{I}_{\mathrm{B}} represents the output characteristic. Relationship between \alpha, \beta and \gamma: Emitter current I_{E}=I_{B}+I_{C}……(1) divide by \mathrm{I}_{\mathrm{C}} gives, \frac{\mathrm{I}_{\mathrm{E}}}{\mathrm{I}_{\mathrm{C}}}=\frac{\mathrm{I}_{\mathrm{B}}}{\mathrm{I}_{\mathrm{C}}}+1 or \frac{1}{\alpha}=\frac{1}{\beta}+1 \beta=\frac{\alpha}{1-\alpha} ## Comparison table between CB, CE and CC configuration So, that’s all from this blog, I hope you get the idea about the Working of NPN and PNP transistor. If you enjoyed this explanation then don’t forget to share it with your friends. Also, read What is Transistor To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. What is Transistor and its types – Physics, Electronics – eSaral A Transistor is an electronic device formed by P and N types of semiconductors. If you want to know about what is transistor and its types then read this article till the end. ## Transistor • Transistor is used in place of triode valve, it is an electronic device formed by P and N types of semiconductors. It was discovered by American scientist J Barden, W.H. Bratain, W. Shockley. • The transistor is three-terminal and two P-N Jn. device. ## There are two types of transistors 1. N-P-N transistor 2. P-N-P transistor A transistor (P-N-P or N-P-N) has the following sections: Emitter: It emits the charge carriers and it is heavily doped. Base: The middle section of the transistor is known as the base. This is very lightly doped and very thin (10^{-6} m). Collector: This is moderately doped. It collects the charge carriers Important points • The cross-sectional area of the base is very large as compared to the emitter. • The cross-sectional area of the collector is less than the base but greater than the emitter. • The base is much thinner than the emitter while the collector is wider than both because the emitter emits charge carriers and the collector has to receive a maximum of them. Since the base is very thin, so recombination in the base region is very less maximum 5%. • The emitter is heavily doped so that it can provide a large number of charge carriers (electrons or holes) into the base. The base is lightly doped and is very thin, hence the majority of charge carriers move on to the collector. This lower doping decreases the conductivity (increases the resistance) of the base material by limiting the number of charge carriers to the collector. The collector is moderately doped. • The emitter is always connected in forward biased and the collector is always connected in reverse biased. • The resistance of the emitter-base junction (forward-biased) is very small as compared to a collector-base junction (reverse biased). Therefore, the forward bias applied to the emitter base is generally very small whereas reverse bias on the collector base is much higher. • Arrowhead always shows the emitter’s current direction. • Current conduction within the P-N-P transistor takes place by hole conduction from emitter to collector. • Conduction in the external circuit is carried out by electrons. So, that’s all from this blog. I hope you get the idea about what is transistors and its types. If you have any confusion related to this topic then feel free to ask in the comments section down below. Also, read Different types of PN-junction diodes To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. Different types of PN-junction diodes – Electronics – eSaral Diode is a PN junction device. If you want to know about the different types of PN-junction diodes then keep reading. ## Different types of PN-junction diodes ### 1. Light Emitting Diode (LED) Very much used in a dancing light display of music systems and information boards on railway stations. In a forward-biased diode, the energy produced by recombination of electrons and holes at the junction can be emitted. If the energy is in the visible region, such a diode is called a light-emitting diode or LED. The color of emitted light depends on the type of material used Ga-As Infrared radiation (Invisible) Ga-P Red or green light Ga-As-P Red or yellow light A circuit for LED is shown in fig. The brightness can be controlled by \mathrm{R}_{\mathrm{L}}. ### 2. Photodiode A PN junction diode made of photosensitive semiconductor is called a photodiode. In photodiode, provision is made for allowing the light to fall at the junction. Its function is controlled by the light allowed to fall on it. In a semiconductor, the electrons jump from the valence band to the conduction band by absorbing energy from some external source of energy. Energy of light photon E=\frac{h c}{\lambda} If this energy is sufficient to break a valence bond, when such light falls on the junction, new hole-electron pairs are created. The number of charge carriers increases and hence the conductivity of the junction increases. Fig. shows a circuit in which the photodiode is reverse-biased. The applied voltage is less than the breakdown voltage. When visible light of suitable energy (hn > forbidden energy gap) is made incident on the photodiode, current begins to flow due to shifting of electrons from valence band to conduction band. This current increases with the increase in the intensity of incident light. If the intensity of light increases to a value, the current becomes maximum. This maximum current is called saturation current. ### 3. Solar Cell A pn-junction diode in which one of the P-or N regions is made very thin (so that the light energy is not greatly absorbed before reaching the junction) is used in converting light energy to electrical energy. Such diodes are called solar cells. In the solar cell, the thin region is called the emitter and the other base. When a light incident on the emitter, a current in the resistance R_{\mathrm{L}} (Fig.). The magnitude of the current depends on the intensity of light. Unlike a photodiode, a solar cell is not given any biasing. It supplies emf like an ordinary cell. The solar cell is based on the photovoltaic effect. When the light of suitable frequency is made incident on an open-circuited solar cell, an emf is produced across its terminals. This emf is called photo-voltaic emf, the effect is called the photovoltaic effect. Uses 1. We can use a set of solar cells to charge storage batteries in the daytime. These batteries can be used for power during the night. 2. Solar cells are extensively used in calculators, wristwatches, and light meters (in photography). 3. The power source in artificial satellites is a solar panel which is an array of solar cells. ### 4. Zener Diode Zener diode is a PN junction diode. By careful adjustment of the concentration of acceptor and do not impurity atoms near the junction, the characters beyond the turn over-voltage become almost a vertical line. Thus, in this region of its characteristic curve, the reverse voltage across the diode remains almost constant for a large change of the reverse current. Therefore, a Zener diode is used as a voltage reference device for stabilizing a voltage at a predetermined value. Zener diodes have been designed to operate from 1 to several hundred volts. In the diodes which are operated below 6V the breakdown of the junction is due to the Zener effect. In those operated between 5 and 8V, the breakdown is due to both the Zener effect and the avalanche multiplication. In general, all diodes which are operated in the breakdown region of their reverse characteristic are known as Zener diodes. Zener diode is a reverse-biased heavily doped P-N junction diode. Which is operated in the breakdown region. So, that’s all from this article. I hope you get the idea about the Different types of PN-junction diodes. If you found this Explanation helpful then please share it with your friends and social media followers. Also read Bridge rectifier circuit Diagram To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. Bridge rectifier circuit diagram – Definition, Efficiency – eSaral Bridge Rectifier is also a full-wave rectifier. If you want to learn about the Bridge rectifier and Bridge rectifier circuit diagram then keep reading. ## Bridge Rectifier It is also a full-wave rectifier. Use of bridge rectifier The bridge rectifier is used in the rectifier type voltmeter. The circuit arrangement is shown in fig. The rectifier elements are p-n junction diodes with a sensitive DC. ammeter as a load. This circuit can be used for the measurement of AC as well as DC voltage and currents. The DC ammeter reads the average value of currents. This may be calibrated to give r.m.s. values. The bridge rectifier has the following advantages • No center tap is required in the transformer secondary, hence both half cycles are similar. • PIV across each diode is \mathrm{E}_{0} (which is 2 \mathrm{E}_{0} in a full wave rectifier). • For a given power, output power transformer of small size can be used as the current in both the primary and secondary of the plate supply transformer flows for the entire cycle. The main disadvantages of bridge rectifier are • The circuit requires two extra diodes • It has poor voltage regulation. ## The efficiency of rectifier: The efficiency rectifier is defined as the ratio of DC output power to the AC input power \eta=\frac{\text { dc power delivered to the load }}{\text { ac input power from transformer sec ondary }} =\frac{P_{d c}}{P_{a c}} =\frac{\mathrm{I}^{2} \mathrm{dc} \mathrm{R}_{\mathrm{L}}}{\mathrm{I}_{\mathrm{rms}}^{2}\left(\mathrm{R}_{\mathrm{F}}+\mathrm{R}_{\mathrm{L}}\right)} For bridge rectifier \eta=\frac{0.812 R_{L}}{2 R_{f}+R_{L}} for ideal diode R_{f}=0 \eta=81.2 \% So, that’s all from this article. I hope you get the idea about the Bridge rectifier and Bridge rectifier circuit diagram. If you liked this article then please share it with your friends. Also, read Full Wave Rectifier circuit diagram To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. Full wave rectifier circuit diagram – Definition, Ripple Factor – eSaral A device that rectifies both halves of the ac input is called a full-wave rectifier. If you want to learn about the full-wave rectifier and full-wave rectifier circuit diagram then keep reading. ## Full-wave rectifier: A rectifier that rectifies both halves of the ac input is called a full-wave rectifier. During the first half of the input cycle, the upper end of S coil is at positive potential and the lower end is at the negative potential the junction diode \mathrm{D}_{1} will get forward biased, while the diode \mathrm{D}_{2} reverse biased. The conventional current due to the diode \mathrm{D}_{1} will flow. When the second half of the input cycle comes, the situation will be exactly reverse. Now, the junction diode \mathrm{D}_{2} will conduct and the current will flow. ## The efficiency of rectifier: The efficiency rectifier is defined as the ratio of dc output power to the ac input power \eta=\frac{\text { dc power delivered to the load }}{\text { ac input power from transformer sec ondary }} =\frac{P_{d c}}{P_{a c}} =\frac{\mathrm{I}^{2} \mathrm{dc} \mathrm{R}_{\mathrm{L}}}{\mathrm{I}_{\mathrm{rms}}^{2}\left(\mathrm{R}_{\mathrm{F}}+\mathrm{R}_{\mathrm{L}}\right)} for Full wave rectifier \eta=\frac{.812}{1+\frac{R_{\mathrm{F}}}{R_{\mathrm{L}}}} if \frac{R_{F}}{R_{L}}<<1 \eta=81.2 \% ## Ripple and Ripple factor: AC components are present in rectifier output these are known as ripple and they are measured in a factor which is known as Ripple Factor Total Current Output \mathrm{I}_{\mathrm{rms}}=\sqrt{\mathrm{I}_{\mathrm{ac}}^{2}+\mathrm{I}^{2} \mathrm{dc}} Ripple Factor =r=\frac{I_{\mathrm{ac}}}{\mathrm{I}_{\mathrm{dc}}} r=\sqrt{\left(\frac{I_{r m s}}{I_{d c}}\right)^{2}-1} For full-wave or bridge rectifier \mathrm{I}_{\mathrm{rms}}=\frac{\mathrm{I}_{\mathrm{m}}}{\sqrt{2}} I_{\mathrm{dc}}=\frac{2 \mathrm{I}_{\mathrm{m}}}{\pi} r=0.48 So, that’s all from this article. I hope you get the idea about the full-wave rectifier and full-wave rectifier circuit diagram. If you have any confusion related to this blog then feel free to ask in the comments section down below. Also Read Half Wave Rectifier Circuit Diagram To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. Half wave rectifier circuit diagram – Definition, Explanation – eSaral A device that converts alternating current into Direct current is called a rectifier. If you want to learn about the half-wave rectifier and half-wave rectifier circuit diagram then you are at the right place. ## Application of diode as a rectifier: An electronic device that converts alternating current into Direct current is called a rectifier. ### Half wave rectifier: A rectifier, which rectifies only one half of each ac supply cycle is called a half-wave rectifier. During the first half of the input cycle, the junction diode gets forward bias. The conventional current will flow. The upper end of R_{L} will be positive potential with respect to the lower end during the second half cycle junction diode will get reverse biased and hence no output will be obtained across R_{L}. Input voltage \mathrm{V}_{\mathrm{i}}=\mathrm{V}_{\mathrm{m}} \sin \omega \mathrm{t} \mathrm{i}=\mathrm{I}_{\mathrm{m}} \sin \omega \mathrm{t} for 0 \leq \omega t \leq \pi \mathrm{i}=0 for \pi<\omega t<2 \pi I_{m}=\frac{V_{m}}{R_{f}+R_{L}} here \mathrm{R}_{\mathrm{f}}= forward resistance of diode R_{L} = load resistance a. dc output current : \mathrm{I}_{\mathrm{dc}}=\frac{1}{2 \pi} \int_{0}^{2 \pi} \mathrm{idt} =\frac{1}{2 \pi}\left[\int_{0}^{\pi} \mathrm{I}_{\mathrm{m}} \sin t \mathrm{dt}+\int_{\pi}^{2 \pi} 0 \mathrm{dt}\right] \mathrm{I}_{\mathrm{dc}}=\frac{\mathrm{I}_{\mathrm{m}}}{\pi}=0.318 \mathrm{I}_{\mathrm{m}} b. dc output voltage: V_{d c}=I_{d c} \times R_{L} =\frac{I_{m}}{\pi} \times R_{L} =\frac{\mathrm{V}_{\mathrm{m}}}{\pi\left[1+\left(\mathrm{R}_{\mathrm{f}} / \mathrm{R}_{\mathrm{L}}\right)\right]} V_{d c}=\frac{V_{m}}{\pi}=0.318 \mathrm{~V}_{\mathrm{m}} c. (Root mean square) RMS current: I_{r m s}=\left[\frac{1}{2 \pi} \int_{0}^{2 \pi} i^{2} d(t)\right]^{1 / 2} =\frac{I_{m}}{2} same V_{r m s}=\frac{V_{m}}{2} So, that’s all from this blog. I hope you enjoyed this explanation of the half-wave rectifier and half-wave rectifier circuit diagram. If you liked this article then please share it with your friends. Also Read What is Diode in electronics To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. What is Diode in electronics – Definition, Important Points – eSaral A diode is a PN junction device. If you want to know what is diode in electronics. Then keep reading. ### Diode A diode is a PN junction device ## Ideal diode • Conducts with zero resistance when forward biased. • Offer an infinite resistance when reverse biased. ## Practical Circuit For diode ### The practical characteristic curve for a diode: Important terms related to diode: 1. Knee VoltageKnee voltage is defined as the forward voltage at which the current through the junction starts increasing rapidly.For silicon = 0.7 volt for germanium = 0.3 volt 2. Forward resistance or ac resistance:It is defined as the reciprocal of the slope of the forward characteristic curve.forward resistance r_{f}=\frac{1}{\text { slope of forwardcharacteristic }} =\frac{1}{\Delta \mathrm{I}_{\mathrm{f}} / \Delta \mathrm{V}_{\mathrm{f}}} =\frac{\Delta \mathrm{V}_{\mathrm{f}}}{\Delta \mathrm{l}_{\mathrm{f}}} 3. Junction breakdown :When the reverse voltage is increased a point is reached when the junction breaks down with sudden rise in reverse current. This value of the voltage is known as the breakdown voltage. Two types of breakdown occur:a. Zener breakdown: a. Zener breakdown: Takes place in junction which are heavily doped so having narrow depletion layers. A very strong electric field appears across the narrow depletion layer which breaks the bond. b. Avalanche breakdown : Occur in junctions which are lightly doped. (having wide depletion layer) so at a high electric field, the minority charge carriers, while crossing the junction acquire very high velocities. A chain reaction is established, giving rise to the high current. 4. Diffusion Current :Some electrons and holes have more kinetic energy \left[\frac{1}{2} \mathrm{mv}^{2}>\mathrm{eV}\right] So \mathrm{e}^{-} diffuse from n to p side and hole diffuse from p to n side due to diffusion of the charge carriers a current will flow known as diffusion current. a. Because of the concentration difference diffusion occurs. b. Diffusion results in an electric current from p side to the n side c. When P-N. Jn is in no Bias = diffusion current = drift current Net charge flow = 0 Net current = 0 5. Drift Current: Due to thermal collisions, the covalent bond is broken. If an electron-hole pair is created in the depletion region, there is a regular flow of electrons towards the n side and of holes towards the p side. Current flow n side to p side called drift current.Drift current and the diffusion current are in the opposite direction So, that’s it from this blog. I hope you get the idea about what is a diode in electronics. If you found this Explanation helpful then share it with your friends and followers. Also read Types of Semiconductor To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. What is P-N junction – Definition, Explanation, Types – eSaral P-N junctions are formed by diffusing trivalent impurity to one-half side and pentavalent impurity to another side. If you want to know what is P-N junction then keep reading this article. ## P-N Junction • By merely Joining the two pieces a P-N Junction cannot be formed. • P-N junctions are formed by diffusing trivalent impurity to one-half side and pentavalent impurity to another side. • The plane dividing the two zones is known as a junction. • P-N junction is unohmic • As P-type semiconductor has a high concentration of holes and N-type semiconductor has a high concentration of free electrons. there is a tendency of holes to diffuse over to the N side and electron to the P side. • When the hole diffuses from the P to N side then this will neutralize with free-electron similarly when electron diffuse from the N to P side it will neutralize with the hole. So, a depletion layer is formed near the Jn. ### Depletion layer: There is a barrier near a junction that opposes the flow of charge carrier is known as depletion layer width of the depletion layer is in micrometer order. ### Potential Barrier: Potential developed in depletion layer is called P.B. P-side is at lower potential and N-side is at higher potential. P.B. for Ge 0.3 volt P.B. for Si 0.7 volt Electric field: Electric field due to P.B. E=\frac{V}{d} For Ge E=\frac{0.3}{10^{-6}} =3 \times 10^{5} \mathrm{~V} / \mathrm{m} order \mathrm{E} \approx 10^{5} \mathrm{~V} / \mathrm{m} the direction of E due to P.B. N to P-side ## P-N Junction with forward bias : When the Positive terminal of a battery is connected to the P side and the negative terminal to the N side. Then PN Junction is in forwarding Bias Forward bias reduces the potential barrier. More charge carriers diffuse across the junction. Special Point: • Potential barrier reduces • Width of the depletion layer decreases • P-N junction offers low resistance in forwarding bias. • Forward current flow in a circuit • The forward characteristic curve is shown in the figure. • Forward dynamic resistance r_{f}=\frac{\Delta V_{f}}{\Delta I_{f}} \cong 100 \Omega • Knee or cut in voltageGe 0.3 VSi 0.7 V • Dependence of forward current on bias voltage \mathrm{I}=\mathrm{I}_{0}\left[\mathrm{e}^{\frac{\mathrm{qV}}{\mathrm{kT}}}-1\right]$$\mathrm{e}^{\frac{\mathrm{qV}}{\mathrm{kT}}}>>1$

$I \approx I_{0} e^{\frac{+q V}{k T}}$ (Approximate exponential growth)

I = Forward current

$\mathrm{I}_{0}$ = reverse saturation current

k = Boltzman constant

q = charge of electron

V = forward voltage

T = temperature

## P-N junction with reverse bias:

When the positive terminal of a battery is connected to the N-side and the negative terminal is connected to the P-side. Holes in the P-region are attracted towards the negative terminal and the electrons in the N-region are attracted towards the positive terminal.

Special Point:
• The depletion layer increases for reverse biased.
• Potential barrier increases
• The reverse characteristic curve is shown in figure
• Very little current called reverse saturation current flows due to minority carrier flow.For Silicon = $10^{-9}$ AFor Germanium = $10^{-6}$ A
• In reverse biased condition, junction behaves as a capacitor of few picofarads.
• In reverse biased condition, junction behaves like high resistive material between two regions.
• In reverse biased P-N diode behaves like an insulator.
• Reverse resistance $\mathrm{R}_{\mathrm{B}}=\frac{\Delta \mathrm{V}_{\mathrm{B}}}{\Delta \mathrm{I}_{\mathrm{B}}} \cong 10^{6} \Omega$$\frac{R_{B}}{R_{f}}=10^{3} : 1 for Geand \frac{R_{B}}{R_{f}}=10^{4} : 1 for Si • Dependence of reverse current on bias volt. \mathrm{I}_{\mathrm{r}}=\mathrm{I}_{0}\left[\mathrm{e}^{-\frac{\mathrm{q} \mathrm{V}}{\mathrm{kT}}}-1\right]$$\mathrm{e}^{-\frac{\mathrm{qV}}{\mathrm{kT}}}<<1$$\mathrm{I} \cong-\mathrm{I}_{0} Here \mathrm{I}_{\mathrm{r}} = reverse current \mathrm{I}_{0} = reverse saturation current V = applied voltage q = charge of electron T = temperature in kelvin Special Point • The diffusion current in the p-n junction is greater than the drift current in magnitude if the junction is forward biased. • A hole diffuses from the p-side to the n-side in a p-n junction. This means that a bond is broken on the n-side and the electron freed from the bond jumps to a broken bond on the p-side to complete it. So, that’s all from this article. I hope you get the idea about what is P-N junction. If you enjoyed this article then please leave your thoughts in the comments section down below. Also, read What is Fermi energy level in semiconductors To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. What is Fermi energy level in semiconductors – Electronics – eSaral Hey, Do you want to know what is fermi energy level in semiconductors? If yes. Then keep reading. ## Fermi Energy Level. Fermi energy is the maximum kinetic energy of an electron at 0 K these electrons are called Fermi electrons and energy level is known as Fermi energy level. • It is always found between the conduction band and valance band • Fermi Level is the energy that corresponds to the center of gravity of the conduction electrons and holes weighted according to their energies. • In pure germanium semiconductor, the Fermi level is about halfway in the forbidden gap. • In an n-type semiconductor, the Fermi level lies in the forbidden gap, very close to the conduction band. • In p-type semiconductor, the Fermi level lies in the forbidden gap, very close to the valence band. • With rising in temperature the Fermi level moves towards the center of the forbidden gap, for both types of semiconductors. • An n-type semiconductor is better than a p-type semiconductor as electrons have more mobility than holes. ## Mass Action Law Under thermal equilibrium, the product of concentration \mathrm{n}_{\mathrm{e}} of free electrons and the concentration \mathbf{n}_{\mathrm{h}} of holes is constant & independent of the amount of doping by donor & acceptor impurity. \mathrm{n}_{e} \mathrm{n}_{\mathrm{h}}=\mathbf{n}_{\mathrm{i}}^{2} where \mathrm{n}_{\mathrm{i}}= intrinsic concentration However, the intrinsic concentration is a function of temperature. • In n-type semiconductors, the number density of electrons is nearly equal to the number density of donor atoms \mathrm{N}_{\mathrm{d}} and is very large as compared to the number density of holes. \mathrm{n}_{\mathrm{e}} \approx \mathrm{N}_{\mathrm{D}} N_{D}>>n_{h} • In p-type semiconductor, the number density of holes is nearly equal to the number density of acceptor atoms \mathrm{N}_{\mathrm{a}} and is very large as compared to number density of electrons. \mathbf{n}_{\mathrm{h}} \approx \mathrm{N}_{\mathrm{A}} \mathrm{N}_{\mathrm{A}}>>\mathrm{n}_{\mathrm{e}} So, that’s all from this article. I hope you get the idea about what is fermi energy level in semiconductors. If you found this article informative then please share it with your friends. If you have any confusion related to this topic, then you can ask in the comments section down below. For a better understanding of this chapter, please check the detailed notes of Electronics. To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. Types of Semiconductors – Types, Examples – eSaral Semiconductor in an extremely pure form is known as an intrinsic semiconductor and The impure semiconductor material is called an extrinsic semiconductor. If you want to learn about the Types of Semiconductors then keep reading. ## Types of Semiconductors ### Pure or intrinsic semiconductor: A semiconductor in an extremely pure form is known as an intrinsic semiconductor. In an intrinsic semiconductor, the number of free electrons is always equal to the number of holes when an external field is applied across the intrinsic semiconductor the conduction through the semiconductor is by both free electrons and holes. • Total current I=I_{e}+I_{h} • It is perfectly neutral • Number of free electron = Number of holes \left(n_{e}=n_{h}\right) ### Extrinsic semiconductor: The intrinsic semiconductors are of little importance because of negligible conductivity and moreover, the conductivity has little flexibility. The electrical conductivity of an intrinsic semiconductor is zero at absolute zero and very small at ordinary (room) temperatures. The conductivity is considerably increased by adding some impurity element to the pure (intrinsic) semiconductors. The impure semiconductor material is called an extrinsic semiconductor. The electrical conductivity of extrinsic semiconductors is called extrinsic conductivity. Depending on impurity added, the extrinsic semiconductor can be divided into two parts- N-Type: An n-type Ge is obtained by adding a small quantity, for 10^{6} Ge atoms approximately one impurity atom of a pentavalent impurity added the ratio of Ge and As atoms is 10^{6}: 1. Generally, Arsenic (As) is taken for this purpose. Each arsenic atom replaces one Ge atom at its crystal lattice site without changing its structure. Four of the five valence electrons of As occupy the same positions in four covalent bonds as earlier occupied by four electrons of replaced Ge atom. But the fifth excess electron remains free. The energy level of this excess electron is only slightly smaller than the lowest energy level of the conduction band. Very small energy of about 0.01 eV (0.05 eV for Si) can detach this electron from the impurity atom. The thermal energy at room temperature is sufficient to enable this electron to detach itself and become a member of the conduction band and take part in conduction. Why n-type Electrons with negative charges help in current conduction, the impure Ge is called n-type. Since impurity atoms donate electrons, the impurity is called donor impurity. Each 1 \mathrm{~cm}^{3} of Ge crystal having 4.52 \times 10^{22} Ge atom, will have 4.52 \times 10^{16} (one millionth) As atoms and hence as many free electrons. The conductivity of the crystal is increased considerably. At high temperatures, some covalent bonds are broken. Then more electrons and holes become free. Free holes in n-type Ge act as minority carriers. • \mathrm{n}_{\mathrm{e}}$$>>$$\mathrm{n}_{\mathrm{h}}$

• $\mathrm{I}=\mathrm{I}_{\mathrm{e}}+\mathrm{I}_{\mathrm{h}}$

$\left(\mathrm{I}_{\mathrm{e}}>>\mathrm{I}_{\mathrm{h}}\right)$ $I \approx I_{e}$

In N-Type semiconductor current mainly flow due to free electrons.

• Energy needed to detach fifth electron from impurity for 0.01 eV for Ge, 0.05 eV for Si.

• Electrons are majority carriers due to the addition of pentavalent impurity.

• Holes are minority carriers due to the breaking of covalent bonds.

• N-type semiconductor has an excess of free electrons but it is electrically neutral.

• It is called donor-type impurity because it gives one electron to a crystal.

• The type of conductivity is called negative or N-type conductivity.

#### P-type:

A p-type Ge is obtained by adding a small quantity for $10^{6}$ Ge atoms approximately one impurity atom of a trivalent impurity added. Ratio of Ge and Al atoms is $10^{6}: 1$.
Generally, Aluminum (Al) or Boron (B) is taken for this purpose.

Each aluminum atom replaces one Ge atom at its crystal lattice site without changing its structure. The three valence electrons of Al occupy the same positions in three covalent bounds as earlier occupied by four electrons of replaced Ge atom.

One covalent bond remains unfilled which shown an electron vacancy.

This electron vacancy is called a hole. The energy of this hole is slightly more than the highest energy level of the valence band (figure).

The electrons from the valence band get easily excited by thermal energy at room temperature to enter the hole. But the electrons filling these holes create new holes in the valence band. These holes in the valence band are filled by more electrons and this continues.

Holes can move freely through a crystal lattice and take part in conduction.

Why p-type

Since holes with positive charges help in current conduction, the impure Ge is called p-type. Since impurity atoms accept electrons (for their holes), the impurity is called acceptor impurity. Each $1 \mathrm{~cm}^{3}$ of Ge crystal having $4.52 \times 10^{22}$ Ge many holes. The conductivity of the crystal is increased considerably.

At high temperatures, some covalent bonds are broken. Then more electrons and holes become free.

Free electrons in p-type Ge act as minority carriers.

• $n_{h}>>n_{e}$

• $\mathrm{I}=\mathrm{I}_{\mathrm{e}}+\mathrm{I}_{\mathrm{h}}$

$\left(\mathrm{I}_{\mathrm{h}}>>\mathrm{I}_{\mathrm{e}}\right)$
$\mathrm{I} \approx \mathrm{I}_{\mathrm{h}}$

In p-type semiconductor current mainly flow due to holes.

• In a p-type semiconductor, majority carriers are positive holes due to the addition of trivalent impurity.

• In a p-type semiconductor, minority carriers are electrons due to the breaking of covalent bonds.

• This impurity is called acceptor type impurity

• In P-type, the valence electrons move from one covalent bond to another bond.

For a better understanding of this chapter, please check the detailed notes of Electronics. To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App.

Effect of Temperature on semiconductor – Electronics – eSaral

Hey, do you want to learn about the Effect of Temperature on Semiconductor? If yes. Then keep reading.

## Effect of Temperature on Semiconductor

### At Absolute zero temperature

Covalent Bonds are very strong and there is no free electron in the conduction band. For semiconductors at 0 K, the conduction band is empty while the valence band is full.

### Above Absolute zero temperature

When temperature increases some of the covalent bonds are break due to thermal energy supplied.

At room temperature, due to thermal energy, some electrons jump to the conduction band.

In the conduction band electrons are free to move. If a potential difference is applied across semiconductor crystals, these free electrons constitute an electric current.

The variation in the resistivity of a pure semiconductor is mainly due to a change in carrier concentration.

The number of electrons raised from VB to the CB

$n_{e} \propto e^{-E_{o} / k T}$

$\mathbf{n}_{\mathrm{e}}=\mathrm{AT}^{3 / 2} \mathrm{e}^{-\Delta \mathrm{Eg} / 2 \mathrm{kT}}$

$\mathrm{T}=$ Absolute Temp.

$\mathrm{k}=$ Boltzmann Const. $=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}$ As temperature (T) increases,

$\mathrm{n}_{\mathrm{e}}$ also increases

i.e. conductivity of the semiconductor increases.

As the temperature of the semiconductor increases, its resistivity decreases.

## Hole:

At higher temperatures, some of the electrons gain energy due to thermal agitation and move from the VB to the CB.

The electron leaving the VB to enter CB leave behind an equal number of vacant sites near the top of the VB. These vacant sites are called holes.

These holes represent the vacancy of electrons and behave like a positive charge.

### Characteristics of hole:

• The hole carries a positive charge equal to an electronic charge.

• The energy of a hole is high as compared to that of the electron.

• The mobility of the hole is smaller than that of the electron.

• In the external electric field, holes move in a direction opposite of that of the electron.

• The effective mass of hole > Effective mass of the electron.

### Electron-hole Recombination:

The completion of a bond may not be necessarily due to an electron from a bond of a neighboring atom. The bond may be completed by a conduction band electron i.e., free electron, and known as electron-hole recombination.

The breaking of bonds or generation of electron-hole pairs and completion of bonds due to recombination is taking place all the time.

At equilibrium, the rate of generation becomes equal to the rate of recombination, giving a fixed number of free electrons and holes.

So, that’s all from this article. I hope you get the idea about the Effect of Temperature on Semiconductor. If you found this article informative then please share it with your friends. If you have any confusion related to this topic, then you can ask in the comments section down below.

For a better understanding of this chapter, please check the detailed notes of Electronics. To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App.

Classification of solids in terms of the forbidden energy gap – eSaral

hey, do you want to learn about the Classification of solids in terms of the forbidden energy gap? If yes. Then you are at the right place.

## Classification of solids in terms of the forbidden energy gap

The width of the forbidden energy gap between the valence band and the conduction band distinguishes conductors, semiconductors, and insulators from each other.

## Difference between Valence, Conduction, and Forbidden Band

### Valence band:

This band is never empty. VB may be partially or completely filled with electrons. VB electrons are not capable of gaining energy from the external electric fields. Therefore, the electrons of VB also not contribute to the electric current.

### Conduction band:

Electrons are rarely present. CB either empty or partially filled with electrons. CB electrons can gain energy from the external electric field. Electrons in this band contribute to the electric current.

### Forbidden energy gap:

Electrons are not found in this band. FEB is completely empty. The minimum energy required for shifting electrons from the valence band to the conduction band is called band gap ($\left(E_{g}\right)$).

So, that’s all from this article. I hope you get the idea about the Classification of the solids according to the Forbidden energy gaps. If you found this article informative then please share it with your friends. If you have any confusion related to this topic, then you can ask in the comments section down below.

For a better understanding of this chapter, please check the detailed notes of Electronics. To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App.

Energy band theory in solids – Electronics, Physics – eSaral

Hey, do you want to learn about the Energy band theory in solids? If yes. Then you are at the right place.

## Energy Bands in solid

### Valence band:

The electrons in the outermost shells are not strongly bonded to their nuclei, these electrons are valence electrons. The band formed by these electrons is called the valence band.

### Conduction band:

Valence electrons are loosely attached to the nucleus. Even at ordinary temperature, some of the valence electrons left the valence band. These are called free electrons and also called conduction electrons. The band occupied by these electrons is called the conduction band.

### Forbidden energy gap ($\Delta$Eg.):

The separation between the conduction band and valence band is known as the forbidden energy gap.

$\Delta \mathrm{Eg} .=(\mathrm{CB})_{\mathrm{Min}}-(\mathrm{VB})_{\mathrm{Max}}$

• No free electrons are in F.E.G.

• The width of FEG depends upon the nature of the substance

• Width is more than valence electrons are strongly attached with the nucleus

• The width of FEG represent in eV

• If the conduction band is empty, then-current conduction is not possible.

• The Greater is the energy gap, the more tightly the valence electrons are bounded to the nucleus. The energy gap in some semiconductors is as follows-

The energy gap decreases slightly with an increase in temperature.