# Numerical Problems : Electromagnetic Induction

#### LEVEL – II

Q:1. A conducting bar of sufficient length is pulled with a constant velocity in a conducting < shaped rail as shown in the figure. Inward magnetic field of induction B is present inside the area bounded by the bar & the rail. Find the external power delivered in moving the rail with constant velocity v at time t ( A = area of cross section of the bar, ρ = resistivity of the bar)

Ans: $\displaystyle \frac{2B^2 A v^3 tan\theta}{\rho}$

Q:2. An LR circuit having a time constant of 50 ms is connected with an ideal battery of emf E. Find the time elapsed before

(a) the current reaches half its maximum values,

(b) the magnetic field energy stored in the circuit reaches half its maximum value.

Ans: (a) $\displaystyle \frac{ln2}{20} sec$ (b) 50 x 10-3 ln(0.3) sec

Q:3. A long solenoid that has 800 turns per meter carries a current i = 3 sin (400t) A. Find the electric field inside the solenoid at a distance 2 mm from the solenoid axis. Consider only the field tangential to a circle having its center on the axis of the solenoid.

Ans: 1.2 × 10-3 cos 400t v/m

Q:4. A metallic rod of length l & resistance R is free to rotate about one of its ends over a smooth, rigid circular metallic frame of radius l in an inward magnetic field of induction B. What torque should be applied by an external agent to rotate the rod with constant angular velocity ω ?

Ans: $\displaystyle \frac{B^2 l^4 \omega}{4 R}$

Q:5. A sliding conducting bar of mass m, resistance R is released from rest. It starts sliding due to the current drawn from a battery of emf E, in a steady inward magnetic field. Find the variation of its speed with time. Also find the terminal speed of the bar.

Ans:$\displaystyle \frac{E}{B l}(1- e^{-B^2 l^2 t/mR})$ ; $latex \displaystyle \frac{E}{B l}$

Q:6. An inductor of inductance 20 mH is connected across a charged capacitor of capacitance 5 μF & resulting L-C circuit is set oscillating at its natural frequency. The maximum charge q is 200 μC on the capacitor. Find the potential difference across the inductor, when the charge is 100 μC.

Ans: 20 V

Q:7. The current in the inner coil is I = 2t2. Find the heat developed in the outer coil between t = 0 and t seconds. The resistance of the inner coil is R and take b >> a.

Ans: $\displaystyle \frac{4 \mu_0^2 \pi^2 a^4}{b^2 R } \frac{t^3}{3}$

Q:8. In the figure shown is a R-L circuit connected with a cell of emf E through a key k. If key k is closed find the current drawn by the battery

(a) just after the key k is closed

(b) long after the key k is closed

Ans: (a)$\displaystyle \frac{E}{2 R}$ (b) $latex \displaystyle \frac{3 E}{5 R}$

Q:9. A very small circular loop of area 5 × 10-4 m2, resistance 2 ohm and negligible self inductance initially coplanar and concentric with a much larger fixed circular loop of radius 0.1 m. A constant current of 1.0 A is passed through the bigger loop. The smaller loop is rotated with constant angular velocity ω rad/sec about it’s diameter. Calculate the

(a) induced emf and

(b) the induced current through the smaller loop as a function of time.

Ans: (a) Induced emf = AB sinωt = 3.14 × 10-9sin ωt

(b) Induced current = 1.57 × 10-9 ω sin ωt

Q:10. A wire in the form of a sector of radius l and of angle (θ = π/4) having a resistance R is free to rotate about an axis passing through point O and perpendicular to horizontal plane. A vertical magnetic field B = −Bo k^ exists in the space. If the sector rotates with constant angular velocity so that Q Joules of heat is produced per revolution, find the constant angular velocity.

Ans: $\displaystyle \frac{8 Q R}{\pi B^2 l^4}$

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