Relation between Electric Field, Potential & Potential Energy of a System of Charges | Physics for JAMB PDF Download

Introduction

• 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.
• For a uniform field, the relationship between electric field (E), potential difference between points A and B (Δ), and distance between points A and B (d) is: E = - Δϕ/d
• If the field is not uniform, calculus is required to solve. Potential is a property of the field that describes the action of the field upon an object.

Relation Between Electric Field and Electric Potential

• 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 field—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 (Δ), and distance between points A and B (d) is:
   E = - Δϕ/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  = − dx/dV .
• This can be derived from basic principles. Given that ∆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 ∆P and W with other terms. ∆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 = −qE_xdx.

Potential Energy of A System of Charges

• Consider the charges q₁ and q₂ initially at infinity and determine the work done by an external agency to bring the charges to the given locations.
• Suppose, charge q₁ is brought from infinity to the point r₁. There is no external field against which work needs to be done, so work done in bringing q1 from infinity to r₁ 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 q₁.
• From the definition of potential, work done in bringing charge q₂ from infinity to the point r₂ is q times the potential at r₂ due to q₁:
  
  where r₁₂ is the distance between points 1 and 2.
• If q₁q₂ > 0, Potential energy is positive. For unlike charges (q₁ q₂ < 0), the electrostatic force is attractive.
• Potential energy of a system of three charges q1, qand q located at r₁, r₂, r, respectively. To bring q first from infinity to r₁, no work is required. Next bring q2 from infinity to r₂. 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,