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**Scalar and Vector Potentials**

We want general solution to Maxwell's Equations,

Given what are the fields

From equation (ii) we can still write

⇒

Now from equation (i)

From equation (iv)

Equations (1) and (2) contain all the information of Maxwell’s equations. Thus we need to calculate only four components (one for V and three for instead of six components (three for and three for

**Example 1: In an infinite straight wire constant current I _{0} is turned on abruptly at t = 0 . Then corresponding retarded potentials are given by** .

The electric field is

And the magnetic field is

**Gauge Transformation**

Suppose we have two sets of potentials, which correspond to the same electric and magnetic fields.

Thus,

Since

Two potentials also give the same

So,

Actually, we might as well absorb k (t ) into λ, defining a new λ by adding to the old one. This will not affect the gradient of λ ; it just adds

It follows that

**Conclusion:** For any old scalar function λ , we can add

provided we simultaneously subtract from V. None of these will affect the physical quantities and Such changes in V and are called **gauge transformations.**

**Coulomb and Lorentz Gauge**

Coulomb Gauge reads

Lorentz Gauge condition is

Since

and

Using Lorentz Gauge condition

The virtue of the Lorentz gauge is that it treats V and on an equal footing: the same

differential operator (called the d' Alembertian) occurs in both equations:**Example 2: For a vector potential the divergence of is where Q _{0} is a constant of appropriate dimensions. Find the corresponding scalar potential that makes and V Lorentz gauge invariant.**

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