Question

In a very long solenoid of radius $R$, if the magnetic field changes at the rate of $dB/dt$. $AB=BC$. Calculate the induced emf between the ends of length $AB$, if $AC$ and $BC$ were removed from the circuit.

• A. $R^2\left(\frac{dB}{dt}\right)$
• B. $4R^2\left(\frac{dB}{dt}\right)$
• C. $\frac{1}{2}{R}^2\left(\frac{dB}{dt}\right)$
• D. $2R^2\left(\frac{dB}{dt}\right)$
• E. None of the above

Answer 1

The correct option is C $\frac{1}{2}{R}^2\left(\frac{dB}{dt}\right)$

Join B to O and A to O (imaginary lines). Flux changes across the triangle AOB.

Area of the triangle AOB $A=\frac{1}{2}R\times R=\frac{R^2}{2}$

E.m.f induced between A and B $\xi =A\frac{dB}{dt}$

Thus $\xi =\frac{R^2}{2}\frac{dB}{dt}$