Non-Uniqueness of Electromagnetic Potentials and Gauge Transformations

In terms of electromagnetic potential field vectors are given by
        B = ∇ x A                                    … (1)
and         E = - ∇ V - ∂A/∂t                                … (2)

From equation (1) and (2) it is clear that for a given A and V, each of the field vectors B and E has only one value i.e., A and V determine B and E uniquely. However the converse is not true i.e., field vectors do not determine the potentials A and V completely. This is turn implies that for a given A and V there will be only one E and B while for a give E and B there can be an infinite number of A’s and V’s.

Suppose we have two sets of potentials, (V, A) and (V, A’) which correspond to the same electric and magnetic fields.
Let        A’ = A + α and V’ = V + β

Since the two A’s give the same B, their curls must be equal.
. :        ∇ x α = 0

The curl of the gradient of any scalar vanishes identically. Hence we can write α as the gradient of some scalar λ:
        α = ∇λ
The two potentials also give the same E. So Eq. (2) requires that
        ∇β + ∂α/∂t = 0
Or        ∇ (β + ∂λ/∂t) = 0.

The term in Parentheses is therefore independent of position. However, it could be a function of time, say k(t). Then,
        Β = - ∂λ/λt + k(t)
Actually, we may as well as absorb k(t) into λ, defining a new λ by adding ʃk(t’)dt’ to the old one. This will not affect the gradient of λ. It just adds k (t) to ∂λ/∂t.
. :            V’ = V - ∂λ/∂t
Also            A’ = A + ∇λ
The magnetic Field B remains unchanged by the transformation
            A → A’ = A + ∇λ
Likewise, the electric field E remains unchanged by the transformation
            V → V’ = V - ∂λ/∂t

Here, λ is an arbitrary scalar function. The addition of an arbitrary scalar function to the potential function such that electric and magnetic fields are not affected is known as Gauge transformation. The invariance of fields under gauge transformation is known as Gauge invariance.

Thus we can add ∇λ to A, provided we simultaneously subtract ∂λ/∂t from V. None of this will affect the physical quantities E and B. Such Changes in V and A are called Gauge transformations.