Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Akhilesh Thakur

Physics : Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

The document Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
All you need of Physics at this link: Physics

• The electric scalar and magnetic vector potentials.

• The wave equations for the electromagnetic potentials.

 

Potentials

A potential is a function whose derivative gives a field. Fields are associated with forces; potentials are associated with energy.
The magnetic vector potential Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is defined so that the magnetic field Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is given by:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(1)

The electric scalar potential φ is defined so that the electric field Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is given by:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev( 2 )

Note that in general, the scalar and vector potentials are functions of position and time.
 

Electrostatic Potential

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

The electric field in the presence of a static charge distribution Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is found from Coulomb’s law:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev (3)

where the integral extends over all space. Note that the prime on the coordinates indicates that the coordinate is associated with the charge.

 

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

In terms of the scalar potential, for a static charge distribution, we have:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(4)

Calculating the potential is simpler than calculating the field directly; and one can then use Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev to find the electric field.

Since we have from Maxwells’ equations:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(5)

where

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(6)

it follows that in an homogeneous, isotropic medium:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(7)

and so:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(8)

Equation (8) is called Poisson’s equation.

Equation (4) is the solution to Poisson’s equation, expressed as an integral.

The behaviour of a charged particle in an electric field is determined by the field Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev , rather than by the potential.

SinceScalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev  for an electrostatic field, we can add any function with vanishing gradient to the potential φ, and obtain the same physics. In other words, the behaviour of any electrostatic system is the same under the transformation:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(9)

where Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is a constant (independent of position).

The freedom that we have in choosing the potential is called gauge invariance.
This allows us to choose arbitrarily the point at which  Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Note that if we write the solution to Poisson’s equation (4):

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev (10)

then implicitly (assuming that all charges are within a finite distance from the origin), we make the gauge choice:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(11)

In the presence of sources for the magnetic field (i.e. a current distribution), the magnetic field B can be found from the Biot-Savart law:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev( 1 2 )

where the integral extends over all space.
Generally, the Biot-Savart law is difficult to apply.
It is often easier to first calculate the magnetic vector potential; but first, we need to derive the differential equation for the vector potential.

In a static case (constant fields, charges and currents), the magnetic field is related to the current density by:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(13)

Substituting Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev, and using the vector identity:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(14)

we find:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(15)

This looks like a complicated equation; but there is a way to simplify it...

Suppose that:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(16)

where f is some function of position. Let us define a new vector potential Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(17)

Since:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(18)

for any function Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev the new vector potential Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev gives exactly the same magnetic field as the old vector potential Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev.

However, if we choose Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev such that:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev ( 19)

then:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(20)

In other words, given a vector potential, we can always choose to work with another vector potential that gives the same field as the original one, but that has zero divergence.

Assuming that we make such a choice, then equation (15) for the vector potential becomes:

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(21)

This is again Poisson’s equation - or rather, three Poisson equations, one for each component of the vectors involved.

  Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev( 22 )

 Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

  Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Scalar and Vector Potentials - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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