Potential Energy in an External Field

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Potential Energy of a Single Charge

In the previous chapter, we learnt about specific charge, its location, and its electric field, but here we are considering the potential energy of a charge, in an external field.

Consider an external electric field 'E' produced by an external source, where it can specified or unspecified. But the potential 'V' due to external source has to be specified.

Let us consider charge 'q' does not effect the external source producing the electric field, which implies q is very small. A strong source placed at infinity produces a finite electric field 'E'. But we are concerned in finding the potential of a given charge and not of a source producing the external field.

The electric field 'E' and the potential 'V' may vary from point to point. If the potential at infinity is zero, then work done in bringing a charge 'q' from infinity to the point 'p' in the external field is q.V. This work is stored as potential energy of q. If 'p' has a position vector, 'r' relative to some origin, then potential energy of q at r in an external field is

= qV(r) where V(r) is the external potential at point 'r'.

Potential energy of a system of two charges in an external field

In this context, we have to consider two charges q₁, q₂ located at r₁ and r₂ respectively in an external field E. The work done in bringing charge q₁ from infinity to r₁ is given by q₁ V(r₁). Similarly, the work done in bringing q₂ to r₂, the work done is not only against the external field but also against the field due to q₁.

Work done against the external field = q₂ V (r₂) and

Work done against the field due to charge q₁

where 'r₁₂' is the distance between charge 'q₁' and 'q₂'.

By the superposition principle for field, we add the work done on q₂ against the two fields.

As the path is independent of work, the potential energy of two charges q₁, q₂ located at r₁ and r₂ in an external field is given by

Potential energy of a dipole

Potential energy of a dipole in a uniform external field

In case of a dipole, the charge q₁ = +q and q₂ = -q. Let the external field 'E' be along the x-direction and origin be the centre of a dipole, then the potential energy of the dipole is

The potential difference between q₁ and q₂ equals to the work done in bringing a unit positive charge against the field from q₂ to q₁ and is given by

V(r₁) - V(r₂) = - E . 2a cosq

Negative sign indicates the decrease in potential in the direction of the field. Thus, potential energy of a dipole in a uniform external field 'E' is given by

Then,

V = -q.2a E cosq

V = - p.E

Dipole has minimum potential energy when aligned with the field, and also remember that a dipole in a uniform field experiences a torque (t=p.E).

Thus, if the dipole can fritter away its potential energy, the torque will align the dipole in the direction of the external field, and bringing its potential to a minimum.