Question

$AB$ is a resistanceless conducting rod which forms a diameter of a conducting ring of radius $r$ rotating in a uniform magnetic field $B$ as shown in figure. The resistors $R_1$ and $R_2$ do not rotate. Then the current through the resistor $R_1$

• A. $\frac{B\omega r^2}{2R_1}$
• B. $\frac{B\omega r^2}{2R_2}$
• C. $\frac{B\omega r^2}{2R_1{R}_2}\left(R_1{+R}_2\right)$
• D. $\frac{B\omega r^2}{2\left(R_1{+R}_2\right)}$
• E. answer required

Answer 1

Answer: (A)

$\frac{B\omega r^2}{2R_1}$

Emf induced between A and center of circle or B and center of circle

$={\int }_0^{r}B\left(r\omega \right)dr=\frac{1}{2}B\omega r^2$

This is the also the potential difference across resistor $R_1$.

Hence, $\frac{1}{2}B\omega r^2=i\left(R_1\right)$

$⟹i=\frac{B\omega r^2}{2R_1}$