Question

In the figure shown, $R$ is a fixed conducting ring of negligible resistance and radius $a$. $PQ$ is a uniform rod of resistance $r$. It is hinged at the centre of the ring and rotated about this point in clockwise direction with a uniform angular velocity $\omega$. There is a uniform magnetic field of strength $B$ pointing inwards and the resistance ${}^{\prime }{r}^{\prime }$ is stationary.

• A. Current throught stationary resistance $r$ is zero
• B. Current through stationary resistance $r$ is $\frac{2B\omega a^2}{5r}$
• C. Direction of current in stationary resistance $r$ is from centre to circumference
• D. Direction of current in stationary resistance $r$ is from circumference to centre
  

Answer 1

Answer: (B), (D)

The correct options are
B Current through stationary resistance $r$ is $\frac{2B\omega a^2}{5r}$
D Direction of current in stationary resistance $r$ is from circumference to centre


When a rod of length $a$ is rotated with angular velocity of $\omega$ in a perpendicular field of induction $B$, about one end, the emf across its ends is $\frac{1}{2}B\omega a^2$
As rod PQ is rotated about centre, we can consider that emf generated between centre to Q is $\frac{1}{2}B\omega a^2$,
emf generated between centre to P is $\frac{1}{2}B\omega a^2$


The system can be replaced by a circuit as shown



$\Rightarrow 2i=\frac{\frac{1}{2}B\omega a^2}{r+\frac{r}{4}}$
$2i=\frac{2B\omega a^2}{5r}$
$2i$ is the current through $r$ from circumference to the center.