Point charge in a hollow sphere

[Try hard]

Inside an uncharged, perfect conducting hollow sphere of thickness t and radius R, a point charge Q is placed at a distance D from the centre, where D<R - t. Need to derive a simple expression for the field outside the sphere. In a section passing through Q, sketch qualitatively the charge distribution in the sphere. How do I use Gauss' Law to solve this? Please help

 

[Doc Al]

Welcome to PF!

Here's a hint or two. There will be an induced charge on the inner surface of the sphere distributed so that its field (for r > R-t) will just cancel the field due to the charge. There will be an induced charge on the outer surface of the sphere that will distribute itself uniformly.

 

[M98Ranger]

To derive an expression for the electric field outside the hollow sphere, we can use Gauss' Law. Gauss' Law states that the electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space. In this case, we can imagine a spherical Gaussian surface with radius r, centered at the point charge Q.

Since the sphere is uncharged, the electric field inside the sphere must be zero. Therefore, the electric flux through the Gaussian surface must also be zero. This means that the enclosed charge must also be zero.

Now, let's consider the electric field outside the sphere. Since the point charge Q is at a distance D from the center, the electric field at any point outside the sphere can be expressed as:

E = kQ/r^2

where k is the Coulomb's constant and r is the distance from the point charge.

To use Gauss' Law, we need to integrate this expression over the Gaussian surface. Since the electric field is spherically symmetric, the magnitude of the electric field will be the same at every point on the Gaussian surface. Therefore, the integral can be simplified to:

∫E • dA = E ∫dA = E(4πr^2) = kQ

where dA is the differential area element of the Gaussian surface and the integral is taken over the entire surface.

Solving for E, we get:

E = kQ/4πr^2

This is the expression for the electric field outside the hollow sphere. As r approaches infinity, the electric field approaches zero, which makes sense as the point charge becomes less and less influential at larger distances.

To sketch the charge distribution in the sphere, we can imagine a section passing through the point charge Q. Since the sphere is uncharged, the charge distribution will be uniform throughout the sphere. However, since the point charge Q is placed at a distance D from the center, the charge distribution will be slightly shifted towards that side of the sphere.

In summary, to use Gauss' Law to solve for the electric field outside a hollow sphere with a point charge inside, we can imagine a Gaussian surface and use the fact that the electric flux through the surface is equal to the enclosed charge divided by the permittivity of free space. By setting the electric flux to zero inside the uncharged sphere, we can solve for the electric field outside the sphere.