Question

A conducting loop of resistance $R$ and radius $r$ has its centre at the origin in a magnetic field $B$. When it is rotated about y-axis through $90^\circ$, the net charge flown in the coil is directly proportional to

• A. B
• B. R
• C. r
• D. r^2

Answer 1

Answer: (A), (D)

A. $B$, D. $r^2$

Answer 2

The net charge flown is given by $q = -\frac{\Delta \Phi}{R}$. The change in magnetic flux is $\Delta \Phi = \Phi_f - \Phi_i$. Initially, the loop is in the xy-plane and the magnetic field is along the x-axis ($\vec{B} = B\hat{i}$). The area vector is $\vec{A}_i = (\pi r^2)\hat{k}$. Thus, the initial flux is $\Phi_i = \vec{B} \cdot \vec{A}_i = 0$. After rotating $90^\circ$ about the y-axis, the loop is in the yz-plane. The area vector becomes $\vec{A}_f = (\pi r^2)\hat{i}$. The final flux is $\Phi_f = \vec{B} \cdot \vec{A}_f = (B\hat{i}) \cdot ((\pi r^2)\hat{i}) = B\pi r^2$. Therefore, $\Delta \Phi = B\pi r^2$. The net charge flown is $q = -\frac{B\pi r^2}{R}$. The magnitude of the net charge flown is $|q| = \frac{B\pi r^2}{R}$. Thus, $|q|$ is directly proportional to $B$ and $r^2$, and inversely proportional to $R$.