Question

A constant and uniform magnetic field of strength, $B=1T$ is established in a cylindrical region of radius 10 cm. A square conducting wire loop of size 30 cm x 30 cm is taken and moved so that all the magnetic field lines pass through this loop once. The resistance of the loop is 200 $\Omega$. The net charge flowing through the loop is

• A. 157 μC
• B. 200 μC
• C. 250 μC
• D. 300 μC

Answer 1

Answer: (A)

157 μC

Answer 2

The net charge flowing through a loop due to a change in magnetic flux is given by $\Delta Q = \frac{\Delta \Phi_B}{R}$, where $\Delta \Phi_B$ is the change in magnetic flux and $R$ is the loop resistance. The magnetic field is confined to a cylindrical region of radius $R = 0.1 \, m$. The maximum flux that can pass through the loop occurs when the loop completely encloses this cylindrical region. This maximum flux is $\Phi_{B,max} = B \cdot (\pi R^2) = 1 \, T \cdot \pi (0.1 \, m)^2 = 0.01\pi \, Wb$. Interpreting "all the magnetic field lines pass through this loop once" as the loop moving from a state of zero flux to a state where it fully encloses the field, the change in flux is $\Delta \Phi_B = 0.01\pi \, Wb$.

Therefore, $\Delta Q = \frac{0.01\pi \, Wb}{200 \, \Omega} \approx \frac{0.01 \times 3.14}{200} = 0.000157 \, C = 157 \, \mu C$.