Question

A rod lies across frictionless rails in a uniform magnetic field $\overrightarrow{B}$ as shown in figure. The rid moves to the right with speed $V$. In order to make the induced emf in the circuit to be zero, the magnitude of the magnetic field should

• A. not change
• B. increase linearly with time
• C. decrease linearly with time
• D. decrease nonlinearly with time
• E. answer required

Answer 1

The correct option is D decrease nonlinearly with time

Flux(t) = $\varphi \left(t\right)=B\times height\times length$

emf = $\frac{d\varphi }{dt}=\frac{dB}{dt}\times height\times length+B\times height\times \frac{dlength}{dt}=0$

$\Rightarrow \frac{dB}{dt}\times length=B\frac{dlength}{dt}=Bv$

$\Rightarrow \frac{dB}{dt}=\frac{Bv}{length}=\frac{Bv}{l_0-vt}$

$l_0$ - length at time t=0.

We see that for emf to be zero, B changes non linearly with time.