Question

A ring of radius R = 9 m having a linear charge density $\lambda = \frac{\epsilon_0}{\pi}$ moves towards a solid imaginary sphere of radius $\frac{R}{2}$, so that the centre of ring passes through the center of sphere. The axis of the ring is perpendicular to the line joining the centers of the ring and the sphere. The maximum flux through the sphere in this process is

• A. R
• B. 2R
• C. 0
• D. $\frac{R}{2}$

Answer 1

Answer: (A)

R

Answer 2

The problem statement describes a scenario where a ring of radius $R$ and a sphere of radius $R/2$ interact. The ring moves such that its center passes through the center of the sphere. A critical piece of information is that "The axis of the ring is perpendicular to the line joining the centers of the ring and the sphere."

Let the center of the sphere be at the origin $(0,0,0)$. Let the ring move along the x-axis, so its center is at $(x,0,0)$. The line joining the centers is the x-axis. If the axis of the ring is perpendicular to the x-axis, let's assume it's the z-axis. This means the plane of the ring is perpendicular to the z-axis, i.e., it lies in a plane $z = constant$. Since the center of the ring is at $(x,0,0)$, the ring must lie in the plane $z=0$.

The points on the ring are given by $(x + R\cos\phi, R\sin\phi, 0)$ for $0 \le \phi < 2\pi$. The sphere is defined by $x^2 + y^2 + z^2 \le (\frac{R}{2})^2$.

For a point on the ring to be inside the sphere, it must satisfy: $(x + R\cos\phi)^2 + (R\sin\phi)^2 + 0^2 \le (\frac{R}{2})^2$ $x^2 + 2xR\cos\phi + R^2\cos^2\phi + R^2\sin^2\phi \le \frac{R^2}{4}$ $x^2 + 2xR\cos\phi + R^2 \le \frac{R^2}{4}$

We are looking for the maximum flux, which is proportional to the maximum enclosed charge. This inequality must hold for some $\phi$. Consider the case when the center of the ring passes through the center of the sphere, i.e., $x=0$. The inequality becomes: $0^2 + 2(0)R\cos\phi + R^2 \le \frac{R^2}{4}$ $R^2 \le \frac{R^2}{4}$ This inequality ($1 \le \frac{1}{4}$) is false. This means that when the center of the ring is at the center of the sphere, no part of the ring is inside the sphere. The enclosed charge is zero, and thus the flux is zero.

Given the problem statement, the geometrical configuration described leads to zero flux at the point where the centers coincide. If the question implies a non-zero flux, there might be an error in the problem statement or a non-standard interpretation is required.

However, if we consider a common type of problem where the ring is fully enclosed by the sphere to maximize flux, this would require the ring's radius to be smaller than the sphere's radius. Let's assume there is a typo and the radius of the ring is $R/2$ and the radius of the sphere is $R$. In this case, when the center of the ring passes through the center of the sphere, the entire ring is inside the sphere.

Total charge on the ring: $Q_{ring} = \lambda \times 2\pi (\text{radius of ring}) = \frac{\epsilon_0}{\pi} \times 2\pi \left(\frac{R}{2}\right) = \epsilon_0 R$. According to Gauss's law, the electric flux through the sphere is $\Phi_E = \frac{Q_{enc}}{\epsilon_0}$. If the entire ring is enclosed, $Q_{enc} = Q_{ring} = \epsilon_0 R$. Therefore, the maximum flux through the sphere would be $\Phi_E = \frac{\epsilon_0 R}{\epsilon_0} = R$.

Given $R = 9$ m, the maximum flux would be $9$ Vm.