Inside the Solid Sphere

The electrostatic or electric potential is defined as the total work done by an electric charge to move from one position to another in the given electric field. The electric potential is a scalar quantity. In the following article, we will understand the basic concept of electricity and electric charge. We will further establish why the electric potential inside the solid sphere is greater than the electric potential on the sphere’s surface.

Electric Field

The electric field is defined as a field or area around charged particles in space, the particles in this field experience forces of attraction and repulsion depending on the character of their respective electric charges. 

Let us understand the electric field with the following derivation.

Consider a point charge Q, placed at an origin point O. Also, consider a point charge q at a different point P. Let us assume that the distance between point P and origin point O is r,

Then,

OP = r, 

Then, according to Coulomb’s law, we know that the electric charge Q at origin point O will apply a force on the point charge q at point P. If the point charge q is removed from P, we see that the charge Q will produce an electric field in the surrounding area. 

The electric field produced by the charge Q at point O to at a point r in the surrounding area is given as,

E(r) = (1/4πε0.Q/r2)rˆ.

Here, vector |r| is a unit vector from the origin point towards the direction of the point r.

Electrostatic Potential

The electrostatic potential or the electric potential is defined as the total amount of work done to move a point charge in an electric field from one point to another. 

Let us understand the concept of electrostatic potential by the following derivation. Consider a system with a point charge. 

If the test charge is q, the potential energy of the charge is defined in terms of the work done. 

The work done will be proportional to q and the amount of force applied at any point will be qE, where E is the electric field at that point in the electric charge system. 

The total work done per unit test charge q is characteristic of the electric field associated with the electric field system. This gives us the explanation for the electrostatic potential V for a point charge. In the considered electric field system, let’s assume the point charge was moved from a point P to point R.

Work done by an external force is given by, 

Vp – Vr = (Up – Ur) / q,

Electric Potential Inside the Solid Sphere

To understand the importance of the electric potential inside the solid sphere, let us derive an expression for the potential. 

Consider a large spherical conducting sphere of radius R and a charge Q is applied on this sphere. Let us assume that the charge is uniformly spread throughout the sphere. 

As we know, the total electric field outside the sphere is equal to a point charge Q at the centre of the sphere, while the field inside the sphere becomes zero. So the electrostatic potential outside the sphere is a point charge and inside the sphere, it is a constant if the sphere’s radius is R. 

We thus have,

The electrostatic potential inside the sphere with an electric charge Q is given by a constant,

 = 1 / 4πε0.Q / R.

Consider that a small sphere with a small charge q and radius r is included inside the sphere, and place it at the centre. 

Potential due to small sphere will be given by,

= 1 / 4πε0.(q / r).

For a large shell of radius R,

= 1 / 4πε0.q / R.

Taking both electric charges of the larger and lesser radius, q and Q, into account the total potential V and the potential difference is given by,

V(R) = 1 / 4πε0.(Q/R + q/R),

V(r) = 1 / 4πε0.(Q/R + q/r),

The total electrostatic potential is given by,

V(r) – V(R) = 1 / 4πε0 (1/r – 1/R).

Let us consider that the charge q is a positive charge. The amount of charge Q is accumulated on the larger sphere with radius R. Even when it is with a positive charge, the solid inner sphere is always at a higher potential.

The difference,

 V(r) – V(R) = +ve. 

Conclusion

The electric field of an electric charge is defined as the field created by a system on which a point charge experiences forces of attraction and repulsion. The electric potential or electrostatic potential is the amount of total work done in an electric field required for a point charge to shift from one point to another. 

In a uniformly solid sphere, the electric charge inside the sphere is always larger than the electric potential on the surface of the solid sphere. In the above article, we look into the derivation and importance of the electric potential inside the solid surface.