Electrostatic Potential Energy
Like the potential energy of a mass in a gravitational field we can define electrostatic potential energy of a charge in an electrostatic field.
For the sake of simplicity, let us assume that electrostatic field ~Eis due to charge + Q placed at the origin. Let a small test charge + q be brought from a point A to a point B, against the repulsive force on it due to charge + Q we shall assume that the test charge +q is so small that it does not disturb the original configuration of charge + q at the origin further we assume that an external force ~F ext applied is just sufficient to counter the repulsive electric force ~FE on the test charge q so that net force on test charge q is zero and it maces from A to B without any acceleration.
In this situation, work done by external force is evasive of work done by the electric force and gets fully stored in the charge q in the form of its potential energy.
On reaching B, if the external force applied on q were removed, the electric force will take the test charge + q away from source charge + Q. the strode potential energy at B is used to provide kinetic energy to the charge q in such a way that the sum of kinetic and potential energies at every point is conserved.
Work done by external force in moving the test charge + q from A to B is
WAB = ∫~Fexp dr = - ∫~FE ~dr
This work done against electrostatic force gets stored as potential energy.
Note that charge q possesses a certain amount of electrostatic potential energy at every point in the electric field. Therefore, work done in moving the charge q from A to B increases its potential energy by an amount equal to potential energy difference between points B and A
?U = UB – UA = WAB
Hence, we define electric potential energy difference between two points B and A as the minimum work required to be done by an external force in moving without acceleration a test charge q from A to B this is true for electrostatic field of any charge configuration. Note that work done by an electrostatic field in moving a given charge form one point to another depends only on the initial and final points. It does not depend on the path chosen in going from one point to the other.
Eqn.(2) defines potential energy difference in terms of physically meaningful quantity work therefore, it is only the difference of potential energy which is significant. The actual value of potential energy at a point is not relevant. Thus, there is a freedom in choosing the point where potential energy is zero. For a charge distribution of finite extent, we choose zero electrostatic potential energy at infinity. Therefore, when point A is at infinity, we get form (2)
W ∞ B = UB – UA = UB – 0 = UB.
Hence we may define potential energy of a charge q at a point in an electrostatic field due to any charge configuration as the work done by the external force (equal and opposite to electric force) in bringing the charge q from infinity to that point.
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