Electromagnetic Induction Set-1

Q1. A single-turn rectangular conducting loop has width $w=5.0\ \text{cm}$ (side perpendicular to its velocity) and length $l=10.0\ \text{cm}$ (side parallel to its velocity). It moves with constant velocity $v=0.20\ \text{m s}^{-1}$ into a region of uniform magnetic field $B=0.50\ \text{T}$ directed into the page. While the loop is partially inside the field region (leading edge inside, trailing edge outside), the magnitude of the induced emf in the loop is:

Q2. Assertion (A): When a rigid conducting loop is translated with constant velocity inside a spatially uniform magnetic field, no net emf is induced around the loop. Reason (R): For rigid translation with uniform $\mathbf{v}$ and uniform $\mathbf{B}$, the motional emf is $\varepsilon=\displaystyle\oint(\mathbf{v}\times\mathbf{B})\cdot d\mathbf{l}=(\mathbf{v}\times\mathbf{B})\cdot\oint d\mathbf{l}=0$ since $\displaystyle\oint d\mathbf{l}=0$ for a closed loop.

Q3. The magnetic flux $\Phi(t)$ through a single-turn loop varies with time as follows: it increases linearly from $0$ at $t=0$ to $0.03\ \text{Wb}$ at $t=1\ \text{s}$, then decreases linearly to $-0.03\ \text{Wb}$ at $t=3\ \text{s}$, and then returns linearly to $0$ at $t=4\ \text{s}$. At $t=2\ \text{s}$ the magnitude of the induced emf (single turn) is:

Q4. A conducting rod of length $l=0.50\ \text{m}$ slides without friction on two parallel rails separated by distance $l$, forming a closed circuit of resistance $R=2.0\ \Omega$. The rod moves perpendicular to a uniform magnetic field $B=0.40\ \text{T}$. A constant horizontal force $F=5.0\ \text{N}$ is applied to the rod. When it reaches terminal velocity $v_t$ the magnetic braking force balances $F$. The terminal speed $v_t$ is:

Q5. Assertion (A): A straight conducting rod of length $l$ moving with velocity $\mathbf{v}$ perpendicular to a uniform magnetic field $\mathbf{B}$ develops an emf between its ends equal to $\varepsilon=Blv$ even if the rod is isolated (not part of a closed circuit). Reason (R): Charges in the rod experience Lorentz force $q(\mathbf{v}\times\mathbf{B})$ and separate until an internal electric field $E$ balances the magnetic force, producing a potential difference.

Q6. A coil has $N=200$ turns and area $A=4.0\ \text{cm}^2$. It rotates with angular speed $\omega=150\ \text{rad s}^{-1}$ in a uniform magnetic field $B=0.12\ \text{T}$ about an axis perpendicular to $\mathbf{B}$. The maximum (peak) emf induced in the coil is:

Q7. A small cylindrical bar magnet is dropped through a long vertical copper tube and reaches a terminal velocity due to eddy-current braking. Which of the following modifications will increase the terminal speed of the magnet (i.e., reduce braking)?
(i) Cutting the tube along its length to create a longitudinal slit (breaking circular currents).
(ii) Coating the inner surface of the tube with an insulating layer so induced currents are reduced.
(iii) Replacing the copper tube by an aluminium tube of the same wall thickness.
(iv) Increasing the wall thickness of the copper tube (same conductivity).

Q8. The induced emf across a coil (connected to a resistor $R=3.0\ \Omega$) varies with time as a symmetric triangle: it rises linearly from $0$ at $t=0$ to $6.0\ \text{V}$ at $t=2.0\ \text{s}$ and falls linearly to $0$ at $t=4.0\ \text{s}$. The total electric charge that flows through the resistor from $t=0$ to $t=4.0\ \text{s}$ is:

Q9. Assertion (A): For a coil of fixed geometry placed in a time-varying magnetic field (sinusoidal flux of fixed amplitude), the magnitude of the induced emf in the coil increases with the frequency of the field variation. Reason (R): The coil's electrical impedance $Z=\sqrt{R^2+(\omega L)^2}$ increases with frequency, causing a larger emf across the coil at higher frequency.

Q10. A conducting rod of length $l=0.25\ \text{m}$ slides to the right on parallel rails at constant speed $v=0.20\ \text{m s}^{-1}$. At time $t=0$ the rod's position measured from the left edge of a uniform magnetic region is $x_0=0.10\ \text{m}$, and the magnetic field in the region is increasing uniformly with time: $B(t)=0.30+0.02t\ \text{T}$ (with $t$ in seconds). At time $t=3.0\ \text{s}$ the rod is at position $x= x_0+vt$. Using the flux rate $\dfrac{d\Phi}{dt}=l\big(B\,v + x\,\dfrac{dB}{dt}\big)$, the magnitude of the induced emf in the single-turn loop at $t=3.0\ \text{s}$ is approximately: