Electrostatic potential energy for collection of point charges

Electrostatic potential energy for collection of point charges

The electric potential at a point at a distance r from point charge q₁ is given by

This potential V is the work done to bring a unit positive charge from infinity to the point. Now if the charge q₂ is brought from infinity to that point at a distance r from q₁, the work done is the product of q₂ and the electric potential at that point. Thus we have

W = q₂V

This work done is stored as the electrostatic potential energy U of a system of charges q₁ and q₂ separated by a distance r. Thus we have

The electrostatic potential energy depends only on the distance between the two point charges. In fact, the expression (1.45) is derived by assuming that q₁ is fixed and q₂ is brought from infinity. The equation (1.45) holds true when q₂ is fixed and q₁ is brought from infinity or both q₁ and q₂ are simultaneously brought from infinity to a distance r between them.

Three charges are arranged in the following configuration as shown in Figure 1.30.

To calculate the total electrostatic potential energy, we use the following procedure. We bring all the charges one by one and arrange them according to the configuration as shown in Figure 1.30.

(i) Bringing a charge q₁ from infinity to the point A requires no work, because there are no other charges already present in the vicinity of charge q₁.

(ii) To bring the second charge q₂ to the point B, work must be done against the electric field created by the charge q₁. So the work done on the charge q₂ is W = q₂ V _1B. Here V_1B is the electrostatic potential due to the charge q₁ at point B.

Note that the expression is same when q₂ is brought first and then q₁ later.

(iii) Similarly to bring the charge q₃ to the point C, work has to be done against the total electric field due to both charges q₁ and q₂. So the work done to bring the charge q₃ is = q₃ (V_1C + V_2C). Here V_1C is the electrostatic potential due to charge q₁ at point C and V_2C is the electrostatic potential due to charge q₂ at point C.

The electrostatic potential is

(iv) Adding equations (1.46) and (1.47), the total electrostatic potential energy for the system of three charges q_1, q₂ and q₃ is

Note that this stored potential energy U is equal to the total external work done to assemble the three charges at the given locations. The expression (1.48) is same if the charges are brought to their positions in any other order. Since the Coulomb force is a conservative force, the electrostatic potential energy is independent of the manner in which the configuration of charges is arrived at.

EXAMPLE 1.15

Four charges are arranged at the corners of the square PQRS of side a as shown in the figure.(a) Find the work required to assemble these charges in the given configuration. (b) Suppose a charge q′ is brought to the center of the square, by keeping the four charges fixed at the corners, how much extra work is required for this?

Solution

(a) The work done to arrange the charges in the corners of the square is independent of the way they are arranged. We can follow any order.

(i) First, the charge +q is brought to the corner P. This requires no work since no charge is already present, W_P = 0

(ii) Work required to bring the charge –q to the corner Q = (-q) x potential at a point Q due to +q located at a point P.

W_Q = −q × 1/4πε . q/ = − 1/4πε  .  q²/a

 (iii) Work required to bring the charge +q to the corner R= q × potential at the point R due to charges at the point P and Q.

(iv) Work required to bring the fourth charge –q at the position S = q × potential at the point S due the all the three charges at the point P, Q and R

(b) Work required to bring the charge q′ to the center of the square = q′ × potential at the center point O due to all the four charges in the four corners

The potential created by the two +q charges are canceled by the potential created by the -q charges which are located in the opposite corners. Therefore the net electric potential at the center O due to all the charges in the corners is zero.

Hence no work is required to bring any charge to the point O. Physically this implies that if any charge q′ when brought close to O, then it moves to the point O without any external force.