Potential Energy of Charges in External Electric Field

Potential energy is the stored form of energy within a body when force is applied to a body such as gravitational force or spring force in order to make it work and move from its original place. Potential energy can be expressed in multiple ways such as:

• When there is no electric field,
• When there is a single charge in an external electric field,
• When there are two charges in an external electric field and with presence of a dipole in an external electric field. 

Key Takeaways: Potential energy, Dipole, External electric field, Electric field, Conservative force, Gravitational force, Spring force, Force

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What is Potential Energy?

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• Potential Energy is the energy possessed by a body due to its position or it is the energy which can be stored by the body.
• It arises from the configuration of the system on shifting the body from the reference point and is the ability of doing work by or against a conservative force.
• Potential energy can be positive or negative and is only defined for a conservative force.
• Potential energy depends upon the frame of reference but change in potential energy does not depend upon the frame of reference.

Potential Energy

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What is Electric Field?

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• Electric field is the region of space surrounding the electrically charged objects/particles where a force is exerted by other electrically charged objects/particles.
• When a charged object/particle interacts with another charged object/particle, then that attractive or repulsive interaction is known as electric force.
• The force on a unit positive charge at a point in an electric field is equal to the electric field intensity at that point.

Electric Field

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Potential Energy when there is no External Electric Field

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Consider two positive charges q1 and q2, where q1 is lying near to a point ‘P’ and q2 is at infinity.

We assume that the potential energy at infinity is zero

We know that two like charges repel each other and when we try to bring q2 at point P, q1 and q2 will repel each other and therefore some work will be done against the repulsive force and this work will be stored as the potential energy in q₂.

Work done will be given by,

W = Vq₂

Also,

U =\(\frac{q_1q_2}{r_{12}} \frac{1}{4\pi \epsilon_o}\)

Now we are concerned with potential energy of a charge (s) in an external field.

The external electric field is not produced by the given charge whose potential energy we wish to calculate.

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Potential Energy of a Single Charge in an External Electric Field

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Work done in bringing a charge ‘q’ from infinity to a point in the external field is qV

W=qV_r

Also the work done work is stored in the form of potential energy where work is given by-;

U= qV_r

Potential energy of the System of Two Charges in External Electric Field

Consider two points, r₁ and r₂ in an external electric field and we want to bring charges q₁ and q₂ to the points r₁ and r₂ respectively.

Then, the work done to bring charge q₁ at r₁ will be given by,

U₁= q₁Vr₁

The work done to bring charge q₂ at r₂ will be given by,

U2= q₂Vr₂

But as q₁ and q₂ are like charges so there will be some repulsion between q₁ and q₂, hence some work will be done to bring the charges at r₁ and r₂, which will be stored in the form of potential energy, given by 

U₁₂ = \(\frac{q_1q_2}{4\pi \epsilon_o}\)

Total potential energy will be given by -;

U = U₁ + U₂ + U

U = q₁V_r1 + q₂V_r2 + \(\frac{q_1q_2}{4\pi \epsilon_o}\)

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Potential Energy of a Dipole in an External Electric Field

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An electric dipole is a system formed by two equal and opposite charges placed at a short distance apart.

The product of one charge and the distance between the charges is known as the ‘electric dipole moment.’

It is denoted by ‘P’

P = q×2l

Now, let us consider a dipole with q1 and q2.

Consider charge q1 to be positive and charge q2 to be negative. 

Let this dipole be placed in a uniform electric field ‘E’.

We know that when a charge is placed in an electric field, some force is applied.

As, q1 is a positive charge, force will be applied on the charge q1 in the direction of the electric field, given by,

F= q₁E

As, q₂ is a negative charge, force will be applied on the charge q₂ in the direction opposite to that of the electric field, given by,

F = – q₂E

Therefore, the net force applied will be equal to zero.

But, as the forces are being applied in the opposite directions and also there is some distance between q₁ and q2, there will be some rotating effect in the dipole or we can say there will be torque in the dipole, given by-;

τ = P×E

τ = PE sinθ

When the electric dipole is placed parallel to the electric field then torque will be 0 because the angle between the electric field and the dipole will be 0^o.

Suppose an external torque \(\tau_{ex}\)is applied in such a manner that it just neutralises this torque and rotates it in the plane of paper from θ₁ to θ₂, at an infinitesimal angular speed and without angular acceleration.

The amount of the work done by the external torque will be given by,

W = \(\int^{\theta_1}_{\theta_o} \tau_{ex}.d\theta\)

W = \(\int^{\theta_1}_{\theta_o} PEsin \theta.d\theta\)

W =\( PE\int^{\theta_1}_{\theta_o}sin \theta.d\theta\)

W = PE [-cosθ]^θ?_θ?

W = PE [cosθ?1- cosθ2]

This work is stored as the potential energy of the system.

U_θ = PE (cosθ1- cosθ2)

Let us assume that the initial angle is 90^o.

U_θ = – PEcosθ

Therefore, the potential energy in the external field in an electric dipole at an angle θ, is given by –PEcosθ.

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Things to Remember

• E is produced by sources external to the given charges.
• The external field E is not produced by the given charge(s) whose potential energy we wish to calculate.
• The V electrical potential is a scalar without any direction and E electric field is a vector.
• A body must possess some form of energy in order to contain potential energy.
• An object will have more gravitational potential when it is heavier and higher above the ground level.

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