Electromagnetic Induction Set-3

Q1. A rectangular conducting loop of width $w=5\ \text{cm}$ (side perpendicular to the direction of motion) is being pulled out of a region of uniform magnetic field $B=0.25\ \text{T}$ at constant speed $v=4.0\ \text{m/s}$. The plane of the loop is perpendicular to the field so the flux through the portion inside the region decreases at rate $w\,v$. What is the magnitude of the induced emf in the loop at that instant?

Q2. A coil of $100$ turns has magnetic flux through each turn (in mWb) varying with time as
for $0\le t<2\ \text{s}:\ \Phi(t)=3t$,
for $2\le t<5\ \text{s}:\ \Phi(t)=6$,
for $5\le t\le 8\ \text{s}:\ \Phi(t)=6-2(t-5)$.
What are the magnitudes of induced emf in the coil at $t=1\ \text{s}$ and $t=6\ \text{s}$ respectively? (Note: $1\ \text{mWb}=10^{-3}\ \text{Wb}$)

Q3. A long solenoid of radius $R$ produces a spatially uniform magnetic field $B(t)$ inside while the external field is negligible. Using the Maxwell–Faraday relation, the magnitude of the induced azimuthal electric field $E_\phi(r)$ (tangential to a circle of radius $r$ centered on the axis) depends on $r$. Which option correctly gives the dependence of $|E_\phi(r)|$ on $r$ (in terms of $|dB/dt|$)?

Q4. Assertion (A): A straight conducting rod moving with constant velocity perpendicular to a uniform magnetic field develops a potential difference between its ends.
Reason (R): The potential difference arises because the magnetic flux through the rod changes with time.

Q5. Two parallel conducting rails separated by $l=0.30\ \text{m}$ lie in a uniform magnetic field $B=0.50\ \text{T}$ perpendicular to their plane. A conducting rod of length $l$ slides on the rails with constant speed $v=2.0\ \text{m/s}$ while the rails are connected through a resistor $R=2.0\ \Omega$. Neglecting self‑inductance, the mechanical power that must be supplied to keep the rod moving at constant speed is:

Q6. A very long solenoid of radius $a$ has a uniform magnetic field $B(t)$ inside (external field ≈ 0). The field inside is changing at rate $\alpha=\dfrac{dB}{dt}$ (constant). A single-turn circular conducting loop of radius $r$ ($r>a$) is placed coaxially around the solenoid. The magnitude of the induced emf in the loop is:

Q7. A thin conducting rod of length $L=0.50\ \text{m}$ rotates about one end with angular speed $\omega=100\ \text{rad/s}$ in a plane perpendicular to a uniform magnetic field $B=0.40\ \text{T}$. What is the emf between the pivot (axis) and the free end of the rod?

Q8. A circular coil of radius $r=0.10\ \text{m}$ has $N=200$ turns and resistance $R=10\ \Omega$. A magnetic field perpendicular to its plane varies as $B(t)=0.02\cos(500t)\ \text{T}$. The amplitude of the induced current in the coil is:

Q9. Assertion (A): The induced emf in a closed conducting loop equals $-\dfrac{d\Phi}{dt}$, where $\Phi$ is the magnetic flux through the loop.
Reason (R): An emf is induced in a moving conductor because charges in it experience magnetic Lorentz force $q(\mathbf{v}\times\mathbf{B})$.

Q10. A circular coil has $N=50$ turns, radius $r=0.10\ \text{m}$ and resistance $R=10\ \Omega$. The magnetic field perpendicular to the coil varies as $B(t)=k\,t^{2}$ with $k=0.5\ \text{T/s}^{2}$. How much energy is dissipated as heat in the coil between $t=0$ and $t=1\ \text{s}$?