Suppose a point charge is located at the center of a spherical surface. The electric field at the surface of the sphere and the total flux through the sphere are determined. Now the radius of the sphere is halved. What happens to the flux through the sphere and the magnitude of the electric field

The flux through the sphere will remain the same, and the magnitude of the electric field will increase by four times.

Explanation:

The electric flux is the number of electric field, passing through a given area. It is proportional to the electric field strength and the area through which this field passes.

If the radius of the sphere is halved, the area of the sphere will reduce by square of the reduction, which will be four times. The electric field lines will become closer together, or technically increase by a fourth of its initial value. The resultant effect is that the electric flux will remain the same.

If originally,

Φ = EA cos∅

where Φ is the electric flux through the sphere

E is the electric field on the sphere

A is the area of the sphere.

If the area of the sphere is reduced to half, then,

the area reduces to A/4,

and the electric field increases to be 4E on the sphere.

The flux now becomes

Φ = 4E x A/4 cos∅

which reduces to

Φ = EA cos∅

which is the initial electric flux on the sphere.