Electromagnetic Induction Set-5

Q1. A straight conducting rod of length $0.20\ \mathrm{m}$ moves with speed $4.0\ \mathrm{m\,s^{-1}}$ such that its length remains perpendicular to its velocity. The rod moves through a uniform magnetic field $B=0.30\ \mathrm{T}$; the velocity makes an angle $30^\circ$ with the magnetic field. Using $\mathcal{E}=Blv\sin\theta$, the magnitude of the induced emf between the ends of the rod is

Q2. A conducting rod of length $0.20\ \mathrm{m}$ slides without friction on two parallel rails connected at one end by a resistor $R=2.0\ \Omega$. The assembly lies in a uniform magnetic field $B=0.50\ \mathrm{T}$ directed into the page. The rod is pulled to the right with constant speed $v=0.50\ \mathrm{m\,s^{-1}}$. Calculate the magnitude of the external force required to maintain constant speed (use $\mathcal{E}=Blv$, $I=\mathcal{E}/R$ and $F=IBl$).

Q3. A circular conducting loop of radius $r=0.05\ \mathrm{m}$ and resistance $R=2.0\ \Omega$ is lying in a uniform magnetic field $B=0.60\ \mathrm{T}$ directed into the page. The loop's radius is increasing at the rate $dr/dt=4.0\times10^{-3}\ \mathrm{m\,s^{-1}}$. At the given instant, the magnitude and direction (as seen by an observer facing the plane of the loop) of the induced current are

Q4. Assertion: For a single-turn circular conducting loop of area $A$, resistance $R$ and self-inductance $L$ placed in a uniform magnetic field $B(t)=B_0+at$ (with constant $a$), if the loop initially has zero current then for $t\gg L/R$ the induced current tends to $I=-\dfrac{A\,a}{R}$.
Reason: The induced emf is $-\dfrac{d\Phi}{dt}=-A\,a$, and in steady state $dI/dt\to0$ so the inductive term $L\,\dfrac{dI}{dt}$ vanishes; hence Ohm's law gives $I=-\dfrac{A\,a}{R}$.

Q5. A conducting disc of radius $R=0.10\ \mathrm{m}$ rotates about its central axis with angular speed $\omega=50\ \mathrm{rad\,s^{-1}}$ in a uniform magnetic field $B=0.40\ \mathrm{T}$ perpendicular to the disc. A sliding contact at the center and another at the rim pick off an emf between center and rim. The magnitude of the emf is given by $\mathcal{E}=\dfrac{1}{2}B\omega R^{2}$. Its numerical value is

Q6. A straight conducting rod of length $0.80\ \mathrm{m}$ moves with speed $3.00\ \mathrm{m\,s^{-1}}$ perpendicular to a uniform magnetic field of magnitude $0.50\ \mathrm{T}$. The magnitude of the potential difference between its ends is (use motional emf $\mathcal{E}=B\ell v$):

Q7. A conducting rod of length $L=0.50\ \mathrm{m}$ slides without friction on parallel conducting rails connected by a resistor $R=2.00\ \Omega$, forming a rectangular loop in a region of uniform magnetic field $B=0.40\ \mathrm{T}$ (field into the page). The rod is pulled to the right with constant speed $v=0.20\ \mathrm{m\,s^{-1}}$. Assuming steady motion, the magnitude of the external force required to keep the rod moving at constant speed is:

Q8. A single-turn rectangular coil of area $A=2.0\times10^{-2}\ \mathrm{m^2}$ rotates with constant angular speed $\omega=100\ \mathrm{rad\,s^{-1}}$ about an axis perpendicular to a uniform magnetic field $B=0.20\ \mathrm{T}$. The induced emf is $\varepsilon(t)=NBA\omega\sin(\omega t)$ with $N=1$. The rms value of the induced emf is:

Q9. A series circuit with resistance $R$ and inductance $L$ (time constant $\tau=L/R$) is connected to a DC battery at $t=0$. The current is $I(t)=\dfrac{V}{R}\big(1-e^{-t/\tau}\big)$. Find the time $t$ (expressed in terms of $\tau$) at which the energy dissipated in the resistor from $t=0$ to $t$ equals the energy stored in the inductor at time $t$.

Q10. A rectangular conducting loop of dimensions $a\times b$ and resistance $R$ lies entirely inside a uniform magnetic field $B$ (field normal to the plane). The loop is pulled out of the field region until it is completely outside. The magnitude of the total electric charge that flows around the loop during the entire process is:

Q11. A conducting rod of length $L=0.40\ \text{m}$ slides with constant speed $v=5.0\ \text{m s}^{-1}$ on frictionless parallel rails in a region of uniform magnetic field $B=0.30\ \text{T}$ directed into the page. The rod is perpendicular to the rails and completes a closed circuit with a resistor. Neglect resistance of the rod and rails. The magnitude of the motional emf induced across the rod is

Q12. A rectangular loop of height $h$ (along $y$) and width $w$ (along $x$) moves in the $+x$ direction with constant speed $v$ into the region $x>0$ where the magnetic field perpendicular to the loop plane varies as $B(x)=k\,x$ for $x>0$ and is zero for $x\le 0$. At $t=0$ the leading edge of the loop is at $x=0$. For times $0<t<\dfrac{w}{v}$ (i.e., while the loop is partially inside the region) the magnitude of the induced emf in the loop is

Q13. A conducting rod of length $L=0.20\ \mathrm{m}$ slides without friction on parallel rails (forming a closed circuit) connected to a resistor $R=4.0\ \Omega$. A uniform magnetic field $B=0.50\ \mathrm{T}$ is perpendicular to the plane. A constant external horizontal force $F=0.100\ \mathrm{N}$ is applied and the rod moves with steady speed $v$. Neglect resistance of the rails and rod. The steady speed $v$ is

Q14. Assertion (A): For a rectangular coil of area $A$ rotating about a fixed axis with angular acceleration $\alpha$ (so its angular speed $\omega$ changes in time), the instantaneous induced emf in the coil is proportional to $\alpha$.
Reason (R): If the coil makes angle $\theta(t)$ with the field, the induced emf is $\mathcal{E}=-\dfrac{d}{dt}(B A\cos\theta)=B A\,\omega\sin\theta$, which depends on the instantaneous angular speed $\omega$ (not on $\alpha$).

Q15. A conducting rod of mass $m$ and length $L$ slides without friction on horizontal rails forming a closed circuit of total resistance $R$ in a region of uniform magnetic field $B$ (into the page). At $t=0$ the rod has speed $v_{0}$ and there is no external force thereafter. The magnetic damping causes the rod's speed to decay. The time dependence of the rod's speed is