(a) Use Gauss's law to prove that the electric field outside any spherically symmetric charge distribution is the same as if all of the charge were concentrated into a point charge. (b) Now use Gauss's law to prove that the electric field inside a spherically symmetric charge distribution is zero if none of the charge is at a distance from the center less than that of the point where we determine the field.

First, let's consider a point P outside a spherically symmetric charge distribution with total charge Q. We want to find the electric field at point P. To do this, we will define a Gaussian surface that encloses the charge distribution and passes through point P. The Gaussian surface will be a sphere with center O and radius r, where r is the distance from O to P. Now, we apply Gauss's law to this Gaussian surface: Φ_E = ∮(E*da) = Q_enclosed/ε_0 Since the charge distribution is spherically symmetric, the electric field E has the same magnitude at every point on the Gaussian surface, so we can pull E out of the integral: E∮da = Q_enclosed/ε_0 Integrating both sides over the surface area of the sphere, we have: E(4πr^2) = Q_enclosed/ε_0 Now, we can solve for E by dividing both sides by 4πr²: E = Q_enclosed/(4πε_0*r²) This formula for E is the same as the formula for the electric field outside a point charge with charge Q. Therefore, we have proven that the electric field outside any spherically symmetric charge distribution is the same as if all of the charge were concentrated into a point charge.

Now let's consider a point P' inside the spherically symmetric charge distribution such that no charge is at a distance less than the distance from the center O to P'. We will define a Gaussian surface that passes through point P' and is a sphere with center O and radius r', where r' is the distance from O to P'. Applying Gauss's law to this Gaussian surface, we have: Φ_E = ∮(E*da) = Q_enclosed/ε_0 Since there is no charge enclosed by the Gaussian surface (as per the problem statement), Q_enclosed = 0. As a result, we have: Φ_E = 0 Since the electric flux through the Gaussian surface is zero, the electric field E inside this spherically symmetric charge distribution must also be zero. We have proven that the electric field inside a spherically symmetric charge distribution is zero if none of the charge is at a distance from the center less than that of the point where we determine the field.