Electric Potential Energy and Relation between Electric Field, Potential & Potential Energy | Physics Class 12 - NEET PDF Download

In general, potential energy can be defined as the capacity for doing work that arises from position or configuration. In the electrical case, a charge will exert a force on any other charge and potential energy arises from any collection of charges. In this document, we will study electrical potential energy and its calculation in detail. 

What Is Electric Potential Energy?

• Imagine you have an electric charge, like a tiny particle with electricity. Now, electric potential energy is like a measure of the stored energy this charge has because of its position.
• The electric potential energy of a charge or a group of charges is determined by how much work an outside force does to bring that charge or group from very, very far away (like infinitely far) to where it is now, without making it speed up.

• Electric potential energy is a scalar quantity and possesses only magnitude and no direction. It is measured in terms of Joules and is denoted by V. It has the dimensional formula of ML²T^-3A^-1.

The electric potential energy of a system of point charges is defined as the work required to assemble this system of charges by bringing them close together, as in the system from an infinite distance.

There are two key elements on which the electric potential energy of an object depends:

• Its own electric charge.
• Its relative position with other electrically charged objects.

Potential Energy of a Single Charge

A charge placed in an electric field possesses potential energy and is measured by the work done in moving the charge from infinity to that point against the electric field. If two charges, q₁ and q₂, are separated by a distance d, the electric potential energy of the system is:

Note: If the distance between the charges is very large i.e., r tends to infinity, then U_e = 0

Example 1: Suppose we have a point charge Q=+5μC located in space. Calculate the electrical potential energy of this single charged particle at a distance of r=2m from it.

Solution: 

Potential Energy of a System of Two Charged Particles

The figure shows two + ve charges q₁ and q₂ separated by a distance r. The electrostatic interaction energy of this system can be expressed as work done in bringing charge q₂ from infinity to the given separation from q₁.
It can be calculated as

[ – ve sign shows that x is decreasing]

If the two charges are of opposite signs, then potential energy will be negative as:

Example 2: Suppose we have two point charges Q₁=+3μC and Q₂=−4μC separated by a distance of d=1m. Calculate the electrical potential energy of this system.

Solution: 

Potential Energy for a System of Charged Particles

When more than two charged particles are there in a system, the interaction energy can be given as the sum of interaction energies of all the different possible pairs of particles. For example, if a system of three particles having charges q₁, q_2, and q₃ is given as shown in figure.
The total interaction energy of this system can be given as:

Example: Suppose we have three point charges Q₁=+2μC, Q₂=−3μC, and Q₃=+4μC arranged at the vertices of an equilateral triangle with sides of length d =1m. Calculate the electrical potential energy of this system.

Solution: 

so, U = -10k

Relation Between Electric Field and Electric Potential

• 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.
• 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,
  

Summary 

• At a point midway between two equal and opposite charges, the electric potential is zero, but the electric field is not zero.
• The electric potential at a point is said to be one volt if one joule of work is done in moving one Coloumb of the charge against the electric field.
• If a negative charge is moved from point A to B, the electric potential of the system increases.
• The reference level used to define electric potential at a point is infinity. It signifies that the force on a test charge is zero at the reference level.
• The surface of the earth is taken to be at zero potential since the earth is so huge that the addition or removal of charge from it will not alter its electrical state.