Electromagnetic Induction Set-2

Q1. A conducting rod of length $L=0.40\ \mathrm{m}$ slides without friction on two parallel conducting rails separated by distance $L$. The rails are connected by a resistor $R=2.0\ \Omega$ to form a closed rectangular circuit. 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 $v=1.0\ \mathrm{m\,s^{-1}}$. The magnitude of the magnetic retarding force on the rod is:

Q2. A square loop of side $a=0.10\ \mathrm{m}$ and resistance $R=1.0\ \Omega$ is being pulled out of a region of uniform magnetic field $B=0.60\ \mathrm{T}$ (into the page) at constant speed $v=0.05\ \mathrm{m\,s^{-1}}$. The side $a$ crosses the boundary so the area inside the field decreases at rate $a\,v$. At the instant when half the loop is inside the field, the magnitude of the induced current in the loop is:

Q3. The magnetic flux $\Phi(t)$ through a single-turn loop varies with time as follows: from $t=0$ to $2.0\ \mathrm{s}$ it increases linearly from $0$ to $4.0\times10^{-5}\ \mathrm{Wb}$; from $t=2.0$ to $5.0\ \mathrm{s}$ it remains constant at $4.0\times10^{-5}\ \mathrm{Wb}$; from $t=5.0$ to $8.0\ \mathrm{s}$ it decreases linearly to $-2.0\times10^{-5}\ \mathrm{Wb}$. The magnitude of the induced emf at $t=6.0\ \mathrm{s}$ is:

Q4. A single-turn circular conducting loop of radius $R=0.10\ \mathrm{m}$ and resistance $R_{\ell}=5.0\ \Omega$ is placed in a magnetic field perpendicular to its plane. The field decays from $B_0=0.60\ \mathrm{T}$ to zero (time-dependence not needed). The total charge that flows around the loop during the entire process is:

Q5. A coil of $N=100$ turns and area $A=2.0\times10^{-3}\ \mathrm{m^2}$ rotates with angular speed $\omega=200\ \mathrm{rad\,s^{-1}}$ in a uniform magnetic field $B=0.020\ \mathrm{T}$ with its axis perpendicular to the field. The amplitude of the induced emf in the coil is:

Q6. Assertion (A): If the magnetic flux $\Phi$ through a loop decreases linearly with time, say $\Phi(t)=\Phi_0-kt$ with $k>0$, the magnitude of the induced emf in the loop remains constant.
Reason (R): The induced emf is proportional to the second time derivative of flux, i.e. $\varepsilon\propto d^{2}\Phi/dt^{2}$.

Q7. The current in a primary coil varies as follows: from $t=0$ to $2.0\ \mathrm{s}$ it increases linearly from $0$ to $10.0\ \mathrm{A}$; from $t=2.0\ \mathrm{s}$ to $5.0\ \mathrm{s}$ it stays at $10.0\ \mathrm{A}$; from $t=5.0\ \mathrm{s}$ to $6.0\ \mathrm{s}$ it decreases linearly to $4.0\ \mathrm{A}$. A secondary coil coaxial with the primary has mutual inductance $M=0.020\ \mathrm{H}$. The magnitude of the induced emf in the secondary at $t=1.0\ \mathrm{s}$ is:

Q8. A conducting disc of radius $R=0.20\ \mathrm{m}$ rotates with angular speed $\omega=100\ \mathrm{rad\,s^{-1}}$ about its axis (perpendicular to the disc). A magnetic field normal to the disc varies with radius as $B(r)=B_0\,(r/R)$ where $B_0=0.80\ \mathrm{T}$ is the field at the rim. The magnitude of the emf between centre and rim is:

Q9. Assertion (A): The electric field induced by a time-varying magnetic field is conservative and can be expressed as the gradient of a scalar potential.
Reason (R): Faraday's law states $\displaystyle \oint_{\mathcal{C}}\mathbf{E}\cdot d\mathbf{\ell}=-\frac{d\Phi}{dt}$, which can be non-zero for changing flux; hence the induced electric field is, in general, non-conservative.

Q10. A conducting rod of length $L=0.50\ \mathrm{m}$ slides without friction on parallel rails forming a closed circuit with resistance $R=2.0\ \Omega$. The rod moves in a uniform magnetic field $B=0.40\ \mathrm{T}$ normal to the loop. Its speed decays as $v(t)=v_0 e^{-\alpha t}$ with $v_0=2.0\ \mathrm{m\,s^{-1}}$ and $\alpha=1.0\ \mathrm{s^{-1}}$. The total energy dissipated as heat in the resistor from $t=0$ to $t=\infty$ is: