Potential & Field Formulation for Time Varying Fields Notes | EduRev

Electricity & Magnetism

IIT JAM : Potential & Field Formulation for Time Varying Fields Notes | EduRev

The document Potential & Field Formulation for Time Varying Fields Notes | EduRev is a part of the IIT JAM Course Electricity & Magnetism.
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Scalar and Vector Potentials

We want general solution to Maxwell's Equations,
Potential & Field Formulation for Time Varying Fields Notes | EduRev

Potential & Field Formulation for Time Varying Fields Notes | EduRev

Given Potential & Field Formulation for Time Varying Fields Notes | EduRev what are the fields Potential & Field Formulation for Time Varying Fields Notes | EduRev

From equation (ii) we can still write
Potential & Field Formulation for Time Varying Fields Notes | EduRev
Potential & Field Formulation for Time Varying Fields Notes | EduRev

Potential & Field Formulation for Time Varying Fields Notes | EduRev

Now from equation (i)
Potential & Field Formulation for Time Varying Fields Notes | EduRev

From equation (iv)
Potential & Field Formulation for Time Varying Fields Notes | EduRev

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 Potential & Field Formulation for Time Varying Fields Notes | EduRev instead of six components (three for Potential & Field Formulation for Time Varying Fields Notes | EduRev and three for Potential & Field Formulation for Time Varying Fields Notes | EduRev


Example 1:  In an infinite straight wire constant current I0 is turned on abruptly at t = 0 . Then corresponding retarded potentials are given byPotential & Field Formulation for Time Varying Fields Notes | EduRev . Find the fields corresponding to these potentials.

The electric field is
Potential & Field Formulation for Time Varying Fields Notes | EduRev
And the magnetic field is
Potential & Field Formulation for Time Varying Fields Notes | EduRev

Gauge Transformation

Suppose we have two sets of potentials, Potential & Field Formulation for Time Varying Fields Notes | EduRev which correspond to the same electric and magnetic fields.

Thus,

Potential & Field Formulation for Time Varying Fields Notes | EduRev
Since
Potential & Field Formulation for Time Varying Fields Notes | EduRev
Two potentials also give the same Potential & Field Formulation for Time Varying Fields Notes | EduRev
So,
Potential & Field Formulation for Time Varying Fields Notes | EduRev
Potential & Field Formulation for Time Varying Fields Notes | EduRev
Actually, we might as well absorb k (t ) into λ, defining a new λ by adding Potential & Field Formulation for Time Varying Fields Notes | EduRev to the old one. This will not affect the gradient of λ ; it just adds Potential & Field Formulation for Time Varying Fields Notes | EduRev

It follows that
Potential & Field Formulation for Time Varying Fields Notes | EduRev

Conclusion: For any old scalar function λ , we can add Potential & Field Formulation for Time Varying Fields Notes | EduRev 

provided we simultaneously subtract Potential & Field Formulation for Time Varying Fields Notes | EduRev from V. None of these will affect the physical quantities Potential & Field Formulation for Time Varying Fields Notes | EduRev and Potential & Field Formulation for Time Varying Fields Notes | EduRev Such changes in V and Potential & Field Formulation for Time Varying Fields Notes | EduRev are called gauge transformations. 

Coulomb and Lorentz Gauge

Coulomb Gauge reads
Potential & Field Formulation for Time Varying Fields Notes | EduRev
Lorentz Gauge condition is
Potential & Field Formulation for Time Varying Fields Notes | EduRev
Since
Potential & Field Formulation for Time Varying Fields Notes | EduRev
and
Potential & Field Formulation for Time Varying Fields Notes | EduRev
Using Lorentz Gauge condition
Potential & Field Formulation for Time Varying Fields Notes | EduRev
The virtue of the Lorentz gauge is that it treats V and Potential & Field Formulation for Time Varying Fields Notes | EduRev on an equal footing: the same 

differential operator  Potential & Field Formulation for Time Varying Fields Notes | EduRev (called the d' Alembertian) occurs in both equations:
Potential & Field Formulation for Time Varying Fields Notes | EduRev
Potential & Field Formulation for Time Varying Fields Notes | EduRev

Example 2: For a vector potential Potential & Field Formulation for Time Varying Fields Notes | EduRev the divergence of Potential & Field Formulation for Time Varying Fields Notes | EduRev is Potential & Field Formulation for Time Varying Fields Notes | EduRev where Q0 is a constant of appropriate dimensions. Find the corresponding scalar potential Potential & Field Formulation for Time Varying Fields Notes | EduRev that makes Potential & Field Formulation for Time Varying Fields Notes | EduRev and V Lorentz gauge invariant.

Potential & Field Formulation for Time Varying Fields Notes | EduRev

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