Surface Charge Density of a Sphere

When a sphere has a surface charge density, it means that there is a distribution of electric charge on its surface. The surface charge density, denoted by sigma (σ), is defined as the amount of charge per unit area on the surface of the sphere.

Surface Charge Density of the Shell

To determine the surface charge density of the spherical shell surrounding the sphere, we need to understand the behavior of electric charges on conductive surfaces.

When a charged conductor is in electrostatic equilibrium, the electric field inside the conductor is zero. This means that the charges on the conductor distribute themselves in such a way that the net electric field inside is canceled out.

Gauss' Law and Electric Flux

To understand the distribution of charges on the spherical shell, we can apply Gauss' Law. Gauss' Law states that the electric flux through any closed surface is equal to the charge enclosed divided by the permittivity of the medium.

The electric flux through a closed surface can be calculated by integrating the electric field over the surface. In the case of a conductive shell, the electric field inside the shell is zero. Therefore, the electric flux through the inner surface of the shell is also zero.

Charge Distribution on the Shell

Since the electric flux through the inner surface of the shell is zero, the charge enclosed by the inner surface must also be zero. This means that all the charges on the sphere are distributed on the outer surface of the shell.

Since the charge on the sphere is distributed uniformly, the charge on the shell will also be uniformly distributed. Therefore, the surface charge density of the shell will be the same as that of the sphere.

Conclusion

In conclusion, when a sphere with surface charge density sigma is surrounded by a conductive spherical shell, the surface charge density of the shell will be equal to sigma. This is due to the behavior of electric charges on conductive surfaces and the principle of electric field cancellation inside a conductor. The charge on the sphere distributes itself on the outer surface of the shell, resulting in a uniform surface charge density for both the sphere and the shell.