Potential Energy of Charges in an Electric Field

What is Potential Energy?

Potential energy is the energy that results from the position/configuration of a body. It can come from its position relative to others, internal stress, electrical charge, or its condition. Owing to its position in a gravitational field (gravitational potential energy), an electric field (electrical potential energy), or a magnetic field (magnetic potential energy), an object may have the capacity to do work. A stretched spring or other elastic deformation can have elastic potential strength.

Potential Energy Function

If a force acting on an object is only a function of position, it is said to be a conservative force and can be represented by a potential energy function that satisfies the derivative condition for a one-dimensional case. 

Fx = −dU/dx

After integrating this equation,

U(x) = - ∫ F(x)dx

This is the definition of potential energy. There is an arbitrary integration constant in the above equation, which shows that any constant can be added to the potential energy equation. This means that you can set the potential energy to zero at any point, which is convenient. 

Zero Potential

The nature of potential is such that the zero potential is arbitrary; it can be set as the center of a coordinate system or at the infinity to the system. This is not to suggest it's insignificant. Once the potential zero is set, then every possible value is evaluated with respect to that zero. The zero of electrical potential has been set for convenience, but the choice of this zero point is generally based on some physical or geometrical logic. It is appropriate to set the zero point at infinity for a single point charge or the localized array of charges. But for an infinite line charge, that's not a rational option, since local values of the potential will go to infinity. The earth or ground potential is usually assumed to be zero for functional electrical circuits, and everything is referred to as the earth.

The Potential Energy of a Single Charge in an Electric Field

Let us consider a magnitude q charge put in a magnitude E, which is very small. The electrical potential at a point is equal to the potential electrical energy (Joules) of any charged particle at that position, divided by the particle charge (Coulombs). Here, the E is the external source of the electric field, which might be unknown as well. This external field is not affected by the charge q. In the field, the potential energy of the charge q is equal to the work done to bring the charge from infinity to the point. The potential at infinity is always assumed to be zero; the work performed to bring the charge from infinity to point is assigned as qV. 

The potential energy of a single charge is given by, 

qV(r). 

Where, r is the position vector, and V(r) is external potential at point r. 

The Potential Energy of the System of Two Charges in an Electric Field

Let us consider two charges with magnitude q1  and q2. They are at a distance of r1 and r2 from the origin for the respective charge. Both the charges are placed in the external field of magnitude E. Work done on charge q1 while bringing it from infinity to r1 is q1V(r1), and for the charge, q2 is q2V(r2). Now, in this case, there are two charges, so there will be an electric potential caused due to a charge as well. So for the charge q2, there will be a field due to the q1 along with the electric field E which will be given by

(q1q2)/(4πϵ0r12)

r12 is the distance between the charges q1 and q2.

Now, work done for the charge q2 will be given by: 

q2V(r2) + (q1q2)/(4πϵ0r12)

Now, the total potential energy will be given by: 

q1V(r1) + q2V(r2) + (q1q2)/(4πϵ0r12)

Where, r1 and r2 are the position vector and V(r1), and V(r2) is the external potential at point r1 and r2, respectively.

Fun Facts

Here are some fun facts to learn in relation to potential energy in an external field in class 12:

• The electrical potential V is a scalar and has no direction, whereas the electric field E is a vector.

• To possess potential energy, an entity must be modified by some form of force.

• The heavier an object and the higher it is above the ground, the more energy it has in its gravitational potential.