Point charge in a hollow sphere

In summary, the problem involves a point charge placed inside a hollow conducting sphere. The goal is to derive an expression for the electric field outside the sphere using Gauss' Law. The solution involves considering the induced charge on the inner and outer surfaces of the sphere.
  • #1
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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? :frown: Please help
 
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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.
 
  • #3


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.
 

1. What is a point charge in a hollow sphere?

A point charge in a hollow sphere is a hypothetical concept in which a single positive or negative electrical charge is concentrated at a single point within a hollow sphere. This means that the charge is spread evenly throughout the entire surface of the sphere.

2. How is the electric field calculated for a point charge in a hollow sphere?

The electric field for a point charge in a hollow sphere can be calculated using Coulomb's law, which states that the electric field is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance from the charge.

3. What is the difference between a point charge in a hollow sphere and a solid sphere?

The main difference between a point charge in a hollow sphere and a solid sphere is that in a solid sphere, the charge is distributed evenly throughout the entire volume of the sphere, while in a hollow sphere, the charge is concentrated at a single point within the sphere's surface. This results in different electric fields and potentials for each type of sphere.

4. How does the electric potential change as you move closer to a point charge in a hollow sphere?

As you move closer to a point charge in a hollow sphere, the electric potential increases. This is because the electric potential is directly proportional to the magnitude of the charge and inversely proportional to the distance from the charge. Therefore, as the distance decreases, the potential increases.

5. Can a point charge in a hollow sphere exist in real life?

No, a point charge in a hollow sphere is a theoretical concept and cannot exist in real life. This is because all charges are made up of particles with a finite size and cannot be concentrated at a single point. However, the concept is useful for theoretical calculations and can help understand the behavior of electric fields in certain situations.

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