A thick conducting spherical shell, with inner radius R1 and outer radius R2, is in electrostatic equilibrium. The area inside and outside the spherical shell is vacuum. The net charge on the spherical shell is Q. Note: for parts (a)-(c), describe the direction of the electric field assuming that Q is positive.
(a) Find the electric field outside the shell, at a distance d > R2 from the center of the shell.
(b) Find the electric field in the shell material, at a distance d from the center of the shell where R1 < d < R2.
(c) Find the electric field at a distance d < R1 from the center of the shell, i.e. in the empty space inside the shell.
(d) How much charge is on the inner surface of the conducting shell

Let's use a sphere as a Gaussian surface, in this case the field lines and the radii of the spheres are parallel so the scalar product is reduced to the algebraic product, let's carry out the integral

E A = q_{int} /ε₀

The area of a sphere is

A = 4π r²

we substitute

E (4π r²) =q_{int} /ε₀

a) The field is requested outside the spherical shell

d> R₂

in this case the charge inside the Gaussian surface is

q_{int} = Q

E = Q / (4π r²) ε₀

E = k Q / r²

k = 1 / 4πε₀

b) In the second case, the field inside the spherical casing is requested.

As the surface is metallic, the charge is located on the surface of it, the electrons repel each other. So the charge inside is zero

q_{int} = 0

E = 0

c) The field in the interior region of the shell, all the charge is external, therefore the internal charge is zero

q_{int} = 0

E = 0

d) as the shell is conductive, the electrons can move freely, which is why it moves as far away as possible between them, this means that everything is located as close as possible to the outer surface of the conductor. Consequently, THERE ARE NO ELECTRONS ON THE INTERNAL SURFACE

Q_internal =0