How do I calculate the electric field at the centre of 1/8th part of a hollow sphere having charge ‘Q’ uniformly distributed on its surface. I want to avoid integration.

To calculate the electric field at the center of 1/8th of a hollow sphere with a uniform charge distribution, we can use the principle of superposition and symmetry, which simplifies the problem significantly without the need for integration.

Understanding the Hollow Sphere's Electric Field

First, let's recall that a uniformly charged hollow sphere produces an electric field outside the sphere, but inside the hollow part, the electric field is zero due to symmetry. However, since we are interested in only 1/8th of the sphere, we can think of it as a combination of contributions from the entire sphere and the removal of the other 7/8ths.

Visualizing the Problem

Imagine the hollow sphere as a complete object, and then consider how the electric field behaves at the center of the 1/8th section. The key here is to recognize that the electric field contributions from the charges on the surface of the sphere will cancel out in certain directions due to symmetry.

Using Symmetry and Superposition

For a complete hollow sphere, the electric field inside is zero. However, when we take only 1/8th of the sphere, we can think of it as having a charge density that is effectively concentrated in that section. The electric field at the center of this 1/8th section can be calculated by considering the contributions from the surface charges.

• Let’s denote the total charge on the sphere as Q.
• The surface charge density σ can be expressed as σ = Q / (4πR²), where R is the radius of the sphere.
• For 1/8th of the sphere, the effective charge Q_eff is Q_eff = Q / 8.

Calculating the Electric Field

Now, we can use the fact that the electric field due to a uniformly charged surface at a point just outside the surface is given by:

E = σ / (2ε₀), where ε₀ is the permittivity of free space.

However, since we are interested in the center of the 1/8th section, we need to consider how the electric field vectors from the surface charges combine. The electric field from the entire sphere would point radially outward, but in our case, we only have 1/8th of that contribution.

Thus, the electric field at the center of the 1/8th section can be approximated as:

E_center = (1/8) * (σ / (2ε₀))

Substituting the expression for σ, we get:

E_center = (1/8) * (Q / (4πR² * 2ε₀))

Finally, simplifying this gives:

E_center = Q / (64πε₀R²)

Final Thoughts

This result shows that the electric field at the center of 1/8th of a hollow sphere with charge Q uniformly distributed on its surface is directly proportional to the charge and inversely proportional to the square of the radius of the sphere. This approach avoids complex integration by leveraging symmetry and the properties of electric fields from charged surfaces.