Step 1: Understanding the Concept:
For a solid conducting sphere (or a hollow spherical shell), any excess charge placed on it will instantly repel itself and reside entirely on its outer surface.
Due to this charge distribution, the electrical properties inside the conductor have specific boundary conditions.
Step 2: Key Formula or Approach:
1. Electric Field ($E$): Inside a conductor in electrostatic equilibrium, the electric field is always zero everywhere. $E_{\text{inside}} = 0$.
2. Electric Potential ($V$): Since $E = -\frac{dV}{dr} = 0$, the potential $V$ must be a constant throughout the entire volume of the sphere. This constant value is equal to the potential at the surface of the sphere, which is calculated as if all charge were concentrated at the center: $V = \frac{1}{4\pi\varepsilon_0} \frac{Q}{R}$.
Step 3: Detailed Explanation:
Applying the principles to the center of the sphere (which is inside the conductor):
- The Electric Field at the center is strictly zero. The symmetric distribution of positive charge on the surface pulls equally in all outward directions, canceling the field out perfectly at the center.
- The Electric Potential at the center is not zero. It requires work to bring a test charge from infinity to the surface of the sphere against the repulsion of $Q$. Once on the surface, moving it anywhere inside (including the center) requires no additional work because $E=0$. Thus, the potential at the center is the same as on the surface:
\[ V_{\text{center}} = V_{\text{surface}} = \frac{Q}{4\pi\varepsilon_0 R} \]
Step 4: Final Answer:
The electric field is zero and the electric potential is $\frac{Q}{4\pi\varepsilon_0 R}$.