The document Relation between Field and Potential and Potential Energy of A System of Charges Class 12 Notes | EduRev is a part of the Class 12 Course Physics Class 12.

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**Key Points**

â€¢ The electric field is a measure of force per unit charge; the electric potential is a measure of energy per unit charge.

â€¢ For a uniform field, the relationship between electric field (E), potential difference between points A and B (Î”), and distance between points A and B (d) is: E = - Î”Ï•/d

If the field is not uniform, calculus is required to solve.

â€¢ Potential is a property of the field that describes the action of the field upon an object.**Terms****Electric potential: **The potential energy per unit charge at a point in a static electric field; voltage.**Electric field:** A region of space around a charged particle, or between two voltages; it exerts a force on charged objects in its vicinity.

The relationship between electric potential and field is similar to that between gravitational potential and field in that the potential is a property of the field describing the action of the field upon an object.

**Electric field and potential in one dimension**

The presence of an electric field around the static point charge (large red dot) creates a potential difference, causing the test charge (small red dot) to experience a force and move.

The electric field is like any other vector fieldâ€”it exerts a force based on a stimulus, and has units of force times inverse stimulus. In the case of an electric field the stimulus is charge, and thus the units are NC-1. In other words, the electric field is a measure of force per unit charge.

The electric potential at a point is the quotient of the potential energy of any charged particle at that location divided by the charge of that particle. Its units are JC-1. Thus, the electric potential is a measure of energy per unit charge.

In terms of units, electric potential and charge are closely related. They share a common factor of inverse Coulombs (C-1), while force and energy only differ by a factor of distance (energy is the product of force times distance).

Thus, for a uniform field, the relationship between electric field (E), potential difference between points A and B (Î”), and distance between points A and B (d) is:

E = - Î”Ï•/d

The -1 coefficient arises from repulsion of positive charges: a positive charge will be pushed away from the positively charged plate, and towards a location of higher-voltage.

The above equation is an algebraic relationship for a uniform field. In a more pure sense, without assuming field uniformity, electric field is the gradient of the electric potential in the direction of x:

Ex = âˆ’ dx/dV .

This can be derived from basic principles. Given that âˆ†P=W (change in the energy of a charge equals work done on that charge), an application of the law of conservation of energy, we can replace âˆ†P and W with other terms. âˆ†P can be substituted for its definition as the product of charge (q) and the differential of potential (dV). We can then replace W with its definition as the product of q, electric field (E), and the differential of distance in the x direction (dx):

qdV = âˆ’qE_{x}dx.

**Potential Energy of A System of Charges**

Consider the charges q_{1} and q_{2} initially at infinity and determine the work done by an external agency to bring the charges to the given locations.

Suppose, charge q_{1} is brought from infinity to the point r_{1}. There is no external field against which work needs to be done, so work done in bringing q1 from infinity to r_{1} is zero. This charge produces a potential in space given by

where r1P is the distance of a point P in space from the location of q_{1}.

From the definition of potential, work done in bringing charge q_{2} from infinity to the point r_{2} is q times the potential at r_{2} due to q_{1}:

where r_{12} is the distance between points 1 and 2.

If q_{1}q_{2} > 0, Potential energy is positive. For unlike charges (q_{1} q_{2} < 0), the electrostatic force is attractive.

Potential energy of a system of three charges q1, qand q located at r_{1}, r_{2}, r, respectively. To bring q first from infinity to r_{1}, no work is required. Next bring q2 from infinity to r_{2}. As before, work done in this step is

The total work done in assembling the charges at the given locations is obtained by adding the work done in different steps,

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