Answer :
Final answer:
The potential difference between the center and surface is 12.00 V. Option D is correct.
Explanation:
An insulating sphere with a uniformly distributed charge acts like a point charge located at its center when considering external points. This means the electric field outside the sphere decreases with the inverse square of the distance from the center. However, inside the sphere, the electric field increases linearly with the distance from the center until it reaches its maximum value at the surface.
Therefore, to calculate the potential difference between the center and the surface, we need to integrate the electric field along the path from the center to the surface. This integration gives us:
V(surface) - V(center) = ∫(E * dr)
where:
- V(surface) is the potential at the surface (which we know is 0 for a conductor)
- V(center) is the potential at the center (which we want to find)
- E is the electric field inside the sphere
- dr is the infinitesimal distance element along the path
For a uniformly charged sphere, the electric field inside is given by:
E(r) = Q * r / (4πε₀R³)
where:
- Q is the total charge of the sphere
- r is the distance from the center
- R is the radius of the sphere
- ε₀ is the permittivity of free space
Substituting this into the integration and performing the calculation, we get:
V(center) = Q² / (8πε₀R)
Plugging in the values for Q and R:
V(center) = (4.00 * 10⁻⁶ C)² / (8π * 8.85 * 10⁻¹² F/m * 0.060 m) ≈ 12.00 V (option D).
