The current in a discharging LR circuit without the battery drops from
Question: The current in a discharging LR circuit without the battery drops from $2.0 \mathrm{~A}$ to $1.0 \mathrm{~A}$ in $0.10 \mathrm{~s}$. (a) Find the time constant of the circuit. (b) If the inductance of the circuit is $4.0 \mathrm{H}$, what is its resistance? Solution:...
Read More →Two coils A and B have inductances 1.0H and 2.0H respectively.
Question: Two coils $\mathrm{A}$ and $\mathrm{B}$ have inductances $1.0 \mathrm{H}$ and $2.0 \mathrm{H}$ respectively. The resistance of each coil is $10 \Omega$. Each coil is connected to an ideal battery of emf $2.0 \mathrm{~V}$ at $\mathrm{t}=0$. Let $i_{A}$ and $i_{B}$ be the currents in the two circuit at time t. Find the ratio $i_{A} / i_{B}$ at (a) $\mathrm{t}=100 \mathrm{~ms}$, (b) $\mathrm{t}=200 \mathrm{~ms}$ and (c) $\mathrm{t}=1 \mathrm{~s}$. Solution:...
Read More →Solve this following
Question: Mark $(\sqrt{ })$ against the correct answer in the following: $f y=x^{\sqrt{x}}$ then $\frac{d y}{d x}=?$ A. $\sqrt{x} \cdot x^{(\sqrt{x}-1)}$ B. $\frac{x^{\sqrt{x}} \log x}{2 \sqrt{x}}$ C. $\mathrm{x}^{\sqrt{\mathrm{x}}}\left\{\frac{2+\log \mathrm{x}}{2 \sqrt{\mathrm{x}}}\right\}$ D. none of these Solution:...
Read More →An inductor of inductance 2.00H is joined in series
Question: An inductor of inductance $2.00 \mathrm{H}$ is joined in series with a resistor of resistance $200 \Omega$ and a battery of emf $2.00 \mathrm{~V}$. At $t=10 \mathrm{~ms}$, find (a) the current in the circuit, (b) the power delivered by the battery, (c) the power dissipated in heating the resistor and (d) the rate at which energy is being stored in magnetic field. Solution:...
Read More →Solve this following
Question: Mark $(\sqrt{)}$ against the correct answer in the following: If $y=x^{\sin x}$ then $\frac{d y}{d x}=$ ? A. $(\sin x) \cdot x^{(\sin x-1)}$ B. $(\sin \mathrm{x} \cos \mathrm{x}) \cdot \mathrm{x}^{(\sin \mathrm{x}-1)}$ C. $x^{\sin x}\left\{\frac{\sin x+x \log x \cdot \cos x}{x}\right\}$ D. none of these Solution:...
Read More →Mark against the correct answer in the following:
Question: Mark $(\sqrt{ })$ against the correct answer in the following: If $y=x^{x}$ then $\frac{d y}{d x}=$ ? A. $x^{x} \log x$ B. $\mathrm{x}^{\mathrm{x}}(1+\log \mathrm{x})$ C. $x(1+\log x)$ D. none of these Solution:...
Read More →An LR circuit with emf ε is connected at t=0.
Question: An LR circuit with emf $\varepsilon$ is connected at $\mathrm{t}=0$. (a) Find the charge $Q$ which flows through the battery during 0 to t. (b) Calculate the work done by the battery during this period. (c) Find the heat developed during this period. (d) Find the magnetic field energy stored in the circuit at time t. (e) Verify that the results in the three parts above are consistent with energy conservation. Solution:...
Read More →Solve this following
Question: Mark $(\sqrt{ })$ against the correct answer in the following: If $y=e^{1 / x}$ then $\frac{d y}{d x}=$ ? A. $\frac{1}{\mathrm{x}} \cdot \mathrm{e}^{(1 / \mathrm{x}-1)}$ B. $\frac{-e^{1 / x}}{x^{2}}$ C. $e^{1 / x} \log x$ D. none of these Solution:...
Read More →The magnetic field at a point inside a
Question: The magnetic field at a point inside a $2.0 \mathrm{mH}$ inductor-coil becomes $0.80$ of its maximum value in $20 \mu$ when the inductor is joined to a battery. Find the resistance of the circuit. Solution:...
Read More →A solenoid having inductance
Question: A solenoid having inductance $4.0 \mathrm{H}$ and resistance $10 \Omega$ is connected to a $4.0 \mathrm{~V}$ battery at $\mathrm{t}=0$. Find (a) the time constant, (b) the time elapsed before the current reaches $0.63$ of its steady-state value, (c) the power delivered by the battery at this instant and (d) the power dissipated in Joule heating at this instant. Solution:...
Read More →A coil having an inductance L and a resistance R is
Question: A coil having an inductance $L$ and a resistance $R$ is connected to a battery of emf $\varepsilon$. Find the time taken for the magnetic energy stored in the circuit to change from one fourth of the steady-state value to half of the steady-state value. Solution:...
Read More →An LR circuit having a time constant of
Question: An LR circuit having a time constant of $50 \mathrm{~ms}$ is connected with an ideal battery of emf $\varepsilon$. Find the time elapsed before (a) the current reaches half its maximum value, (b) the power dissipated in heat reaches half its maximum value and (c) the magnetic field energy stored in the circuit reaches half its maximum value. Solution:...
Read More →Figure shows a square frame of wire having a total resistance placed coplanarly with a long,
Question: Figure shows a square frame of wire having a total resistance placed coplanarly with a long, straight wire. The wire carries a current i given by $i=i_{0} \sin \omega t$. Find (a) the flux of the magnetic field through the square frame, (b) the emf induced in the frame and (c) the heat developed in the frame in the time interval 0 to $\frac{20 \pi}{\omega}$. Solution:...
Read More →Consider a situation similar to that of the previous problem except that
Question: Consider a situation similar to that of the previous problem except that the ends of the rod slide on a pair of thick metallic rails laid parallel to the wire. At one end the rails are connected by resistor of resistance R. (a) What force is needed to keep the rod sliding at a constant speed $v$ ? (b) In this situation what is the current in the resistance R? (c) Find the rate of heat developed in the resistor. (d) Find the power delivered by the external agent exerting the force on th...
Read More →A uniform magnetic field B exists in a cylindrical region,
Question: A uniform magnetic field B exists in a cylindrical region, shown dotted in figure. The magnetic field increases at a constant $\frac{d B}{d t}$ rate $^{d t}$. Consider a circle of radius $r$ coaxial with the cylindrical region. (a) Find the magnitude of the electric field $E$ at a point on the circumference of the circle. (b) Consider a point $\mathrm{P}$ on the side of the square circumscribing the circle. Show that the component of the induced electric field at $P$ along ba is the sa...
Read More →A wire of mass m and length l can slide freely on a pair of smooth,
Question: A wire of mass $\mathrm{m}$ and length I can slide freely on a pair of smooth, vertical rails. A magnetic field $\mathrm{B}$ exists in the region in the direction perpendicular to the plane of the rails. The rails are connected at the top end by a capacitor of capacitance $C$. Find the acceleration of the wire neglecting any electric resistance. Solution:...
Read More →Figure shows a situation similar to the previous problem.
Question: Figure shows a situation similar to the previous problem. All parameters are the same except that a battery of emf $\varepsilon$ and a variable resistance $\mathrm{R}$ are connected between $\mathrm{O}$ and $\mathrm{C}$. he connecting wires have zero resistance. No external force is applied on the rod (except gravity, forces by the magnetic field and by the pivot). In what way should the resistance $R$ be changed so that the rod may rotate with uniform angular velocity in the clockwise...
Read More →Consider a variation of the previous problem.
Question: Consider a variation of the previous problem. Suppose the circular loop lies in a vertical plane. The rod has mass $m$. The rod and the loop have negligible resistances but the wire connecting $\mathrm{O}$ and $\mathrm{C}$ has a resistance $\mathrm{R}$. The rod is to made to rotate with a uniform angular velocity $\omega$ in the clockwise direction by applying a force at the midpoint of OA in a direction perpendicular to it. Find the magnitude of this force when the rod makes an angle ...
Read More →Consider the situation shown in the figure of the previous problem.
Question: Consider the situation shown in the figure of the previous problem. Suppose the wire connecting 0 and $C$ has zero resistance but the circular loop has the resistance $\mathrm{R}$ uniformly distributed along its length. The rod $\mathrm{OA}$ is made to rotate with a uniform angular speed $\omega$ as shown in the figure. Find the current in the rod when $\angle \mathrm{AOC}=90^{\circ}$. Solution:...
Read More →Figure shows a conducting circular loop of radius a placed in a uniform,
Question: Figure shows a conducting circular loop of radius a placed in a uniform, perpendicular magnetic field B. A thick metal rod OA is pivoted at the center 0 . The other end of the rod touches the loop at A. The center $O$ and a fixed point $C$ on the loop are connected by a wire OC of resistance R. A force is applied at the middle of the rod OA perpendicularly, so that the rod rotates clockwise at a uniform angular velocity $\omega$. Find the force. Solution:...
Read More →A rectangular metallic loop of length I and width b
Question: A rectangular metallic loop of length I and width b is placed coplanarly with a long wire carrying a current i. The loop is moved perpendicular to the wire with a speed $v$ in the plane containing the wire and the loop. Calculate the emf induced in the loop when the rear end of the loop is at the distance a from the wire. Solve by using faraday's law for the flux through the loop and also by replacing different segments with equivalent batteries. Solution:...
Read More →Figure shows a straight, long wire carrying a current i
Question: Figure shows a straight, long wire carrying a current $i$ and a rod of length I coplanar with the wire and perpendicular to it. The rod moves with a constant velocity $v$ in a direction parallel to the wire. The distance of the wire from the center of the rod Find the motional emf induced in the rod. Solution:...
Read More →The magnetic field in a region is given by
Question: The magnetic field in a region is given by $\vec{B}=\vec{k} \frac{B_{0}}{L} y$ where $\mathrm{L}$ is a fixed length. A conducting rod of length $\mathrm{L}$ lies along the $Y$ axis between the origin and the point $(0, L, 0)$. If the rod moves with a velocity $v=v_{0} \bar{l}$, find the emf induced between the ends of the rod. Solution:...
Read More →Figure shows a conducting disc rotating about its axis in a perpendicular magnetic field B.
Question: Figure shows a conducting disc rotating about its axis in a perpendicular magnetic field $B$. A resistor of resistance $R$ is connected between the center and the rim. Calculate the current in the resistor. Does it enter the disc or leave it at the center? The radius of the disc is $5.0 \mathrm{~cm}$, angular speed $\omega=10 \mathrm{rad} / \mathrm{s}, \mathrm{B}=0.40 \mathrm{~T}$ and $\mathrm{R}=10 \Omega$. Solution:...
Read More →A conducting disc of radius r rotates with a small
Question: A conducting disc of radius $r$ rotates with a small but constant angular velocity $\omega$ about its axis. A uniform magnetic field B exists parallel to the axis of rotation. Find the motional emf between the center and the periphery of the disc. Solution:...
Read More →