Question

A spherical cavity is hollowed out of the interior of aneutral conducting sphere. At the center of the cavity is a pointcharge, of positive charge q.a) What is the total surface charge q_int on the interior surface of the conductor (i.e., onthe wall of the cavity)?b) What is the magnitude E_int of the electric field inside the cavity as a function ofthe distance rfrom the point charge? Let k,as usual, denote \frac{1}{4\pi\epsilon_0} .c) What is the electric field E_ext outside the conductor?

Answer 1

Answer:

A) the total surface charge on the interior surface is equal in module to q+ and oposite in charge ($q_{int}=-q_{(+)}$).

B) The electric field inside the cavity is: $\vec{E_i}(r)=\frac{q_{(+)}\vec{r}}{4\pi\epsilon_0r^2}$.

C) The electric field outside the cavity is: $\vec{E_i}(r)=\frac{q_{(+)}\vec{r}}{4\pi\epsilon_0r^2}$.

Explanation:

The spherical cavity has a neutral charge. This means that this spherical cavity won´t affect the field inside. Therefore the field generated by the charge q+ is the same with or without the spherical cavity.

The total charge of the system can be calculated considering charge conservation:

Qtot0=Qtotf

q(+)=q(+)+qin+qex

qin=-qex and qin+qex=0

Using a Gaussian sphear centered in the charge q(+), we can calculate the field knowing that its direction would be $\vec(r)}$.

If we apply the gaussian sphear between the interior and the exterior radius of the conducting hollow sphear, knowing that the field should be 0:

$\displaystyle\oint_{s} \vec{E}\, \vec{ds}=\displaystyle\oint_{s} \vec{0}\, \vec{ds}=0=\frac{q_{IN}}{\epsilon_0} =\frac{q_{(+)}+q_{int}}{\epsilon_0} \\q_{(+)}+q_{int}=0\\q_{(+)}=-q_{int}$

If we apply the gaussian sphear outside the exterior radius of the conducting hollow sphear, we obtain the field required in c).