Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sin ^{3} x \cos ^{6} x d x$ Solution: Since power of $\sin$ is odd, put $\cos x=t$ Then $d t=-\sin x d x$ Substitute these in above equation, $\int \sin ^{3} x \cos ^{6} x d x=\int \sin x \sin ^{2} x t^{6} d x$ $=\int\left(1-\cos ^{2} \mathrm{x}\right) \mathrm{t}^{6} \sin \mathrm{x} \mathrm{d} \mathrm{x}$ $=\int\left(1-\mathrm{t}^{2}\right) \mathrm{t}^{6} \mathrm{dt}$ $=\int\left(\mathrm{t}^{6}-\mathrm{t}^{8}\right) \mathrm{d} \mathrm{t}$ $=\fra...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sin ^{5} x \cos x d x$ Solution: Let $\sin x=t$ Then $d(\sin x)=d t=\cos x d x$ Put $t=\sin x$ and $d t=\cos x d x$ in above equation $\int \sin ^{5} x \cos x d x=\int t^{5} d t$ $=\frac{t^{6}}{6}+c\left(\right.$ since $\int x^{n} d x=\frac{x^{n+1}}{n+1}+c$ for any $\left.c \neq-1\right)$ $=\frac{\sin ^{6} x}{6}+c$...
Read More →Two charges –q each are fixed separated by distance 2d.
Question: Two charges qeach are fixed separated by distance 2d. A third chargeqof mass m placed at the mid-point is displaced slightly byx(xd) perpendicular to the line joining the two fixed charged as shown in Fig. 1.14. Show thatqwill perform simple harmonic oscillation of time period. $T=\left[\frac{8 \pi^{3} \varepsilon_{0} m d^{3}}{q^{2}}\right]^{1 / 2}$ Solution: Net force F on q towards the centre $\mathrm{O}$ $F=2 \frac{q^{2}}{4 \pi \varepsilon_{\theta} r^{2}} \cos \theta=-\frac{2 q^{2}}...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \cos ^{5} x d x$ Solution: $\int \cos ^{5} x d x=\int \cos ^{3} x \cos ^{2} x d x$ $=\int \cos ^{3} x\left(1-\sin ^{2} x\right) d x\left\{\right.$ since $\left.\sin ^{2} x+\cos ^{2} x=1\right\}$ $=\int\left(\cos ^{3} x-\cos ^{3} x \sin ^{2} x\right) d x$ $=\int\left(\cos x\left(\cos ^{2} x\right)-\cos ^{3} x \sin ^{2} x\right) d x$ $=\int\left(\cos x\left(1-\sin ^{2} x\right)-\cos ^{3} x \sin ^{2} x\right) d x\left\{\right.$ since $\left.\sin ^{2...
Read More →There is another useful system of units,
Question: There is another useful system of units, besides the SI/mks A system, called the cgs (centimeter-gram-second) system. In this system Coloumbs law is given by $\mathbf{F}=\frac{\mathrm{Qq}}{r^{2}} \hat{\boldsymbol{r}}$ where the distanceris measured in cm (= 102m), F in dynes (=105N) and the charges in electrostatic units (es units), where 1es unit of charge = {1/[3]} 10‒9C The number [3] actually arises from the speed of light in vaccum which is now taken to be exactly given by c = 2.9...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sin ^{5} x d x$ Solution: $\int \sin ^{5} x d x=\int \sin ^{3} x \sin ^{2} x d x$ $=\int \sin ^{3} x\left(1-\cos ^{2} x\right) d x\left\{\right.$ since $\left.\sin ^{2} x+\cos ^{2} x=1\right\}$ $=\int\left(\sin ^{3} x-\sin ^{3} x \cos ^{2} x\right) d x$ $=\int\left(\sin x\left(\sin ^{2} x\right)-\sin ^{3} x \cos ^{2} x\right) d x$ $=\int\left(\sin x\left(1-\cos ^{2} x\right)-\sin ^{3} x \cos ^{2} x\right) d x\left\{\right.$ since $\left.\sin ^{2...
Read More →Two fixed, identical conducting plates (α & β),
Question: Two fixed, identical conducting plates ( ), each of surface area S are charged to Qandq, respectively, whereQq 0. A third identical plate (), free to move is located on the other side of the plate with charge q at a distanced(Fig 1.13). The third plate is released and collides with the plate . Assume the collision is elastic and the time of collision is sufficient to redistribute charge amongst (a) Find the electric field acting on the plate before collision. (b) Find the charges on an...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sin ^{4} x \cos ^{3} x d x$ Solution: Let $\sin x=t$ We know the Differentiation of $\sin x=\cos x$ $\mathrm{dt}=\mathrm{d}(\sin \mathrm{x})=\cos \mathrm{xd} \mathrm{x}$ So, $\mathrm{dx}=\frac{\mathrm{dt}}{\cos \mathrm{x}}$ substitute all in above equation, $\int \sin ^{4} x \cos ^{3} x d x=\int t^{4} \cos ^{3} x \frac{d t}{\cos x}$ $=\int t^{4} \cos ^{2} x d t$ $=\int t^{4}\left(1-\sin ^{2} x\right) d t$ $=\int t^{4}\left(1-t^{2}\right) d t$ $=...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \cot ^{6} x d x$ Solution: Let $I=\int \cot ^{6} x d x$ $\Rightarrow I=\int\left(\operatorname{cosec}^{2} x-1\right) \cot ^{4} x d x$ $\Rightarrow I=\int \cot ^{4} x \operatorname{cosec}^{2} x d x-\int \cot ^{4} x d x$ $\Rightarrow I=\int \cot ^{4} x \operatorname{cosec}^{2} x d x-\int\left(\operatorname{cosec}^{2} x-1\right) \cot ^{2} x d x$ $\Rightarrow I=\int \cot ^{4} x \operatorname{cosec}^{2} x d x-\int\left(\operatorname{cosec}^{2} x \cot ...
Read More →Consider a sphere of radius R with charge density distributed as
Question: Consider a sphere of radius R with charge density distributed as (r) =krforrR = 0 forrR. (a) Find the electric field at all points r. (b) Suppose the total charge on the sphere is 2e where e is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution. Solution: (a) The symmetry of the problem suggests that the electric field is radial. For points $rR$, c...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \cot ^{5} x d x$ Solution: Let I $=\int \cot ^{5} x d x$ $\Rightarrow I=\int \cot ^{2} x \cot ^{3} x d x$ $\Rightarrow I=\int\left(\operatorname{cosec}^{2} x-1\right) \cot ^{3} x d x$ $\Rightarrow I=\int \cot ^{3} x \operatorname{cosec}^{2} x d x-\int \cot ^{3} x d x$ $\Rightarrow I=\int \cot ^{3} x \operatorname{cosec}^{2} x d x-\int\left(\operatorname{cosec}^{2} x-1\right) \cot x d x$ $\Rightarrow I=\int \cot ^{3} x \operatorname{cosec}^{2} x d...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \cot ^{5} x \operatorname{cosec}^{4} x d x$ Solution: Let $I=\int \cot ^{5} x \operatorname{cosec}^{4} x d x$ $\Rightarrow I=\int \cot ^{5} x \operatorname{cosec}^{2} x \operatorname{cosec}^{2} x d x$ $\Rightarrow I=\int \cot ^{5} x\left(1+\cot ^{2} x\right) \operatorname{cosec}^{2} x d x$ $\Rightarrow I=\int\left(\cot ^{5} x+\cot ^{7} x\right) \operatorname{cosec}^{2} x d x$ Let $\cot x=t$, then $\Rightarrow-\operatorname{cosec}^{2} x d x=d t$ $...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \cot ^{n} x \operatorname{cosec}^{2} x d x, n \neq-1$ Solution: Let I $=\int \cot ^{n} x \operatorname{cosec}^{2} x d x$ Let $\cot x=t \Rightarrow-\operatorname{cosec}^{2} x d x=d t$ $\Rightarrow I=-\int t^{n} d t$ $\Rightarrow I=-\frac{t^{n+1}}{n+1}+c$ $\Rightarrow I=-\frac{\cot ^{n+1} x}{n+1}+c$ Therefore, $\int \cot ^{n} x \operatorname{cosec}^{2} x d x=-\frac{\cot ^{n+1} x}{n+1}+c$...
Read More →In 1959 Lyttleton and Bondi suggested that the expansion
Question: In 1959 Lyttleton and Bondi suggested that the expansion of the Universe could be explained if matter carried a net charge. Suppose that the Universe is made up of hydrogen atoms with a number density N, which is maintained a constant. Let the charge on the proton be:ep= (1 +y)ewhere e is the electronic charge. (a)Find the critical value of y such that expansion may start. (b)Show that the velocity of expansion is proportional to the distance from the centre. Solution: (a) Suppose univ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \operatorname{cosec}^{4} 3 x d x$ Solution: Let I $=\int \operatorname{cosec}^{4} 3 x d x$ $\Rightarrow I=\int \operatorname{cosec}^{2} 3 x \operatorname{cosec}^{2} 3 x d x$ $\Rightarrow I=\int\left(1+\cot ^{2} 3 x\right) \operatorname{cosec}^{2} 3 x d x$ $\Rightarrow I=\int\left(\operatorname{cosec}^{2} 3 x+\cot ^{2} 3 x \operatorname{cosec}^{2} 3 x\right) d x$ Let $\cot 3 x=t$, then $\Rightarrow-3 \operatorname{cosec}^{2} 3 x d x=d t$ $\Rightar...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sec ^{4} 2 x d x$ Solution: Let I $=\int \sec ^{4} 2 x d x$ $\Rightarrow I=\int \sec ^{2} 2 x \sec ^{2} 2 x d x$ $\Rightarrow I=\int\left(1+\tan ^{2} 2 x\right) \sec ^{2} 2 x d x$ $\Rightarrow I=\int\left(\sec ^{2} 2 x+\tan ^{2} 2 x \sec ^{2} 2 x\right) d x$ Let $\tan 2 x=t$, then $\Rightarrow 2 \sec ^{2} 2 x d x=d t$ $\Rightarrow I=\frac{1}{2} \int\left(1+t^{2}\right) d t$ $\Rightarrow I=\frac{1}{2} t+\frac{1}{2} \cdot \frac{1}{3} t^{3}+c$ $\Ri...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sqrt{\tan x} \sec ^{4} x d x$ Solution: Let $I=\int \sqrt{\tan x} \sec ^{4} x d x$ $\Rightarrow I=\int \sqrt{\tan x} \sec ^{2} x \sec ^{2} x d x$ $\Rightarrow I=\int \sqrt{\tan x}\left(1+\tan ^{2} x\right) \sec ^{2} x d x$ $\Rightarrow I=\int\left(\tan ^{\frac{1}{2}} x+\tan ^{\frac{5}{2}} x\right) \sec ^{2} x d x$ Let $\tan x=t$, then $\Rightarrow \sec ^{2} x d x=d t$ $\Rightarrow I=\int\left(t^{\frac{1}{2}}+t^{\frac{5}{2}}\right) d t$ $\Rightar...
Read More →Five charges, q each are placed at the corners
Question: Five charges, q each are placed at the corners of a regular pentagon of side a (Fig. 1.12). (a) (i) What will be the electric field at O, the centre of the pentagon? (ii) What will be the electric field at O if the charge from one of the corners (say A) is removed? (iii) What will be the electric field at O if the chargeqat A is replaced by q? (b) How would your answer to (a) be affected if pentagon is replaced by n-sided regular polygon with charge q at each of its corners? Solution: ...
Read More →shows the electric field lines around three point charges A,
Question: shows the electric field lines around three point charges A, B and C. (a) Which charges are positive? (b) Which charge has the largest magnitude? Why? (c) In which region or regions of the picture could the electric field be zero? Justify your answer. (i) near A, (ii) near B, (iii) near C, (iv) nowhere. Solution: (a) Charges A and C are positive since lines of force emanate from them. (b) Charge C has the largest magnitude since maximum numbers of field lines are associated with it. (c...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \tan ^{5} x d x$ Solution: Let I $=\int \tan ^{5} x d x$ $\Rightarrow I=\int \tan ^{2} x \tan ^{3} x d x$ $\Rightarrow I=\int\left(\sec ^{2} x-1\right) \tan ^{3} x d x$ $\Rightarrow I=\int \tan ^{3} x \sec ^{2} x d x-\int \tan ^{3} x d x$ $\Rightarrow I=\int \tan ^{3} x \sec ^{2} x d x-\int\left(\sec ^{2} x-1\right) \tan x d x$ $\Rightarrow I=\int \tan ^{3} x \sec ^{2} x d x-\int\left(\sec ^{2} x \tan x\right) d x+\int \tan x d x$ Let $\tan x=t$,...
Read More →Two charges q and –3q are placed fixed
Question: Two chargesqand 3qare placed fixed on x-axis separated by distanced. Where a third charge 2qshould be placed such that it will not experience any force? Solution: The situation given in question is shown in the figure given below: At $P$ : on $2 q$, Force due to $q$ is to the left and that due to $-3 q$ is to the right. $\Rightarrow \frac{2 q^{2}}{4 \pi \varepsilon_{o} x^{2}}=\frac{6 q^{2}}{4 \pi \varepsilon_{o}(d+x)^{2}}$ $\Rightarrow(d+x)^{2}=3 x^{2}$ $\Rightarrow(d+x)=\pm \sqrt{3} x...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sec ^{6} x \tan x d x$ Solution: Let $\mathrm{I}=\int \sec ^{6} \mathrm{x} \tan \mathrm{x} \mathrm{dx}$ $\Rightarrow I=\int \sec ^{5} x(\sec x \tan x) d x$ Substituting, $\sec x=t \Rightarrow \sec x \tan x d x=d t$ $\Rightarrow I=\int t^{5} d t$ $\Rightarrow I=\frac{t^{6}}{6}+c$ $\Rightarrow I=\frac{\sec ^{6} x}{6}+c$ Therefore, $\int \sec ^{5} x(\sec x \tan x) d x=\frac{\sec ^{6} x}{6}+c$...
Read More →represents a crystal unit of cesium chloride,
Question: represents a crystal unit of cesium chloride, CsCl. The cesium atoms, represented by open circles are situated at the corners of a cube of side 0.40 nm, whereas a Clatom is situated at the centre of the cube. The Cs atoms are deficient in one electron while the Clatom carries an excess electron. (i) What is the net electric field on the Clatom due to eight Cs atoms? (ii) Suppose that the Cs atom at the corner A is missing. What is the net force now on the Clatom due to seven remaining ...
Read More →Consider a coin of Example 1.20.
Question: Consider a coin of Example 1.20. It is electrically neutral and contains equal amounts of positive and negative charge of magnitude 34.8 kC. Suppose that these equal charges were concentrated in two point charges separated by (i) 1 cm [ ̴ (1/2) diagonal of the one paisa coin]. (ii) 100 m ( ̴ length of a long building), and (iii) 106m (radius of the earth). Find the force on each such point charge in each of the three cases. What do you conclude from these results? Solution: Suppose,qbe...
Read More →A paisa coin is made up of Al-Mg alloy and weighs 0.75g.
Question: A paisa coin is made up of Al-Mg alloy and weighs 0.75g. It has a square shape and its diagonal measures 17 mm. It is electrically neutral and contains equal amounts of positive and negative charges. Treating the paisa coins made up of only Al, find the magnitude of equal number of positive and negative charges. What conclusion do you draw from this magnitude? Solution: Given, Mass of one paisa coin = 0.75 We know that atomic mass of aluminium = 26.9815 Now, Avogadro's number = 6.023 1...
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