Question

Electric circuit is composed of three conducting rods $\overline { \mathrm { MO } }$, $\overline { \mathrm { ON } }$ &$\overline { \mathrm { PQ } }$ as shown in figure. The resistance of the rods per unit length is known to be λ. The rod $\overline { \mathrm { PQ } }$slides as shown in figure at a constant velocity v keeping it tilt angle relative to $\overline { \mathrm { ON } }$ & $\overline { \mathrm { MO } }$ fixed at 45°. At each resistance the circuit is closed. The whole system is embedded in a uniform magnetic field B, which is directed perpendicularly into the page. Then induced electric current is –

• A. Proportional to time t
• B. Inversely proportional to time t
• C. Proportional to square of time t
• D. Independent of time t

Answer 1

Answer: (D)

Independent of time t

Answer 2

Triangular area at time t

A(t) = $\frac { 2 \mathrm { vt } \times \mathrm { vt } } { 2 }$ = (vt)²

Flux = φ(t) = A(t)B cos 0⁰

φ(t) = (vt)²B

e = = – 2Bv²t

Length of perimeter of triangle = 2vt $\sqrt { 2 }$ + 2vt

= 2vt (1 + $\sqrt { 2 }$)

Electric resistance of circuit

R(t) = 2vt(1 + $\sqrt { 2 }$)λ

Current = =

It is independent of time.