Electric Field at the Center of a Hollow Sphere

To find the electric field at the center of a hollow sphere with a total charge Q distributed uniformly over its surface, we can use Gauss's law. Gauss's law states that the electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the permittivity of the medium.

Gauss's Law:
The electric flux through a closed surface is given by:
Φ = Qenc / ε₀

Where:
Φ is the electric flux through the surface,
Qenc is the total charge enclosed by the surface,
and ε₀ is the permittivity of free space.

Applying Gauss's Law to the Hollow Sphere:

We consider a small area dA on the surface of the hollow sphere. When this area is cut off, the remaining sphere can be divided into two parts: the inner sphere and the outer spherical shell.

1. Electric Flux through the Small Area:
The electric flux through the small area dA is given by:
dΦ = E * dA

Where:
dΦ is the electric flux through the small area,
E is the electric field at the center of the sphere,
and dA is the area of the small surface element.

2. Total Charge Enclosed:
The total charge enclosed by the remaining sphere is Q - dQ, where dQ is the charge on the small area dA.

3. Electric Flux through the Remaining Sphere:
The electric flux through the remaining sphere is given by:
Φ = E * A

Where:
Φ is the electric flux through the remaining sphere,
E is the electric field at the center of the sphere,
and A is the area of the remaining sphere.

4. Applying Gauss's Law:
Using Gauss's law, we can equate the electric flux through the small area to the electric flux through the remaining sphere:
dΦ = Φ

E * dA = E * A

5. Calculating Electric Field at the Center:
Since the electric field is constant at the center of the sphere, we can cancel out the E term:
dA = A

E = Q / (4πε₀ * r²)

Where:
E is the electric field at the center of the sphere,
Q is the total charge on the sphere,
ε₀ is the permittivity of free space,
and r is the radius of the sphere.

Conclusion:
The electric field at the center of a hollow sphere with a total charge Q distributed uniformly over its surface is given by E = Q / (4πε₀ * r²).