Electric Potential of a Charged Sphere

The electric potential at a point is defined as the amount of work done in bringing a unit positive charge from infinity to that point, divided by the magnitude of the charge. In the case of a charged sphere, the electric potential is given by the equation:

V = kQ/r

where V is the electric potential, k is the electrostatic constant, Q is the charge of the sphere, and r is the distance from the center of the sphere.

Determining the Minimum Distance

To find the minimum distance from the surface of the sphere where the electric potential is half of the potential at the center, we need to set up an equation and solve for r.

Let V_c be the electric potential at the center of the sphere, which is equal to V_c = kQ/R, where R is the radius of the sphere. The electric potential at a distance r from the center is given by V = kQ/r.

We can set up the equation:

V = 1/2 * V_c

Substituting the values, we get:

kQ/r = 1/2 * kQ/R

Simplifying the Equation

To simplify the equation, we can cancel out the kQ terms:

1/r = 1/2R

Cross-multiplying, we have:

2R = r

Therefore, the minimum distance from the surface of the sphere where the electric potential is half of the potential at the center is equal to twice the radius of the sphere.

Summary

The minimum distance from the surface of a charged sphere, where the electric potential is half of the potential at the center, is equal to twice the radius of the sphere. This can be derived by setting up an equation using the electric potential formula and solving for the distance. The equation simplifies to 2R = r, where R is the radius of the sphere.