Potential inside the sphere:
When calculating the potential inside the uniformly charged solid sphere, we need to consider the charge distribution within the sphere. The potential at a point inside the sphere can be found using the formula:

V(r) = k * q * (3R^2 - r^2) / (4πε0 * R^3)

Where:
- V(r) is the potential at a distance r from the center of the sphere.
- k is the Coulomb's constant.
- q is the total charge of the sphere.
- R is the radius of the sphere.
- ε0 is the permittivity of free space.

To compute the gradient of V inside the sphere, we need to take the derivative of V with respect to r. Since the potential is a function of r, the gradient of V will give us the electric field inside the sphere.

Potential outside the sphere:
Outside the uniformly charged solid sphere, the potential can be calculated using the formula for a point charge. The potential at a distance r from the center of the sphere is given by:

V(r) = k * q / (4πε0 * r)

To compute the gradient of V outside the sphere, we again take the derivative of V with respect to r. This will give us the electric field outside the sphere.

Gradient of V in each region:
Inside the sphere:
The gradient of V inside the sphere can be found by taking the derivative of the potential formula with respect to r:

dV/dr = -k * q * r / (2πε0 * R^3)

This gives us the electric field inside the sphere, which points towards the center.

Outside the sphere:
The gradient of V outside the sphere can also be found by taking the derivative of the potential formula with respect to r:

dV/dr = -k * q / (4πε0 * r^2)

This gives us the electric field outside the sphere, which follows the inverse square law and points away from the center.

Sketch of V(r):
The sketch of V(r) will have a different shape inside and outside the sphere.

Inside the sphere:
The potential inside the sphere is positive and decreases as we move away from the center. At the center of the sphere, the potential is highest. As we move towards the surface of the sphere, the potential decreases linearly until it reaches zero at the surface.

Outside the sphere:
The potential outside the sphere is negative and also decreases as we move away from the center. At infinity, the potential approaches zero. As we move closer to the sphere, the potential becomes more negative and approaches negative infinity as r approaches the radius of the sphere.

Overall, the potential inside the sphere is positive and decreases linearly, while outside the sphere it is negative and decreases as the inverse of the distance squared.