Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Electromagnetic Theory

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Electrical Engineering (EE) : Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

The document Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Electromagnetic Theory.
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Coulomb Gauge and the Potential formulation of Maxwell’s equations

The Maxwell’s equations in their final form are

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

We had, in the last lecture, made a reformulation of these equations in terms of scalar and vector potentials. This gave us two “coupled” equations for four quantities, i.e. 3 components of the vector potential and one component of scalar potential. We had seen that these equations get decoupled in Lorentz gauge.

We had a lot of discussion on Coulomb gauge in which the divergence of the vector potential is zero. Is this gauge any good to be used now?

Recall the pair of equations,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Note that first equation gets decoupled in Coulomb gauge and becomes a Poisson’s equation for the scalar potential with the formal solution,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

The second equation is not straightforward and requires a lot of mathematical mannipulation before the decoupling can be seen.

To avoid repeating the same expression unnecessarily, we will concentrate on the right hand side of eqn. (2). In Coulomb gauge the term with the divergence drops out and the right hand side of (2) becomes,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

We insert the formal solution of the scalar potential into this equation,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Now the term  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev is a current and we replace it by  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev using the equation of continuity and the term becomes

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

We will do some further simplification to (I). But first let us recall that we have learnt that a vector is completely determined when its divergence and curl are specified. This allows us to write the current density vector inside the integral as a sum of one part  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev whose divergence is zero and another part  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev whose curl is zero,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Using the identity,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

we can write,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

where we have used  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev in arriving at both these relations. We will shortly return to using these relations.

Let us get back to the relation (I). Note that the gradient outside is taken with respect to the point of observation. We can take it inside the integration and it will act only on  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev However, since this depends only on the difference of  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev and  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev we can replace Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev by Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev'by incorporating a minus sign. With this (I) becomes

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

We will simplify this further by using chain rule differentiation,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Substituting this into the preceding term, we have two integrals, one of which is a volume integral of a gradient. This term can be converted to a surface integral like the way we do for the divergence theorem and take to surface to infinity to make this term zero. This implies the remaining integral is over all space and we get

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

in obtaining this step, we have used  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev and used the fact that  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev can be written as  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev because the transverse part has zero divergence. Further, we can use the fact that  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev is irrotational to write,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

With this (I) takes the form

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

At this stage, we will use the Green’s identity for fields T and U according to which

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

As our fields vanish at infinity, we have,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Using this, we get,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Thus the original equation for the vector potential has now been completely decoupled from the scalar part and we have , instead, an inhomogeneous wave equation,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Electromagnetic Momentum

We have seen that the electromagnetic field carries energy. A natural question arises as to whether it carries momentum as well. The answer is affirmative and we will illustrate this by a simple procedure. A more rigorous derivation requires use of the theory of relativity. Suppose, we have two charged particles, q1 and q2, the former moving along the x axis while the latter moves along the y axis.

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

The force on q1 just as it passes by the origin is purely electrical as it lies along the direction of motion of q2, and is given by 

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

where d is the distance between the charges at that instant. This is also the magnitude of the electric force on q2 due to q1. However, if we look at the force on q2 due to q1, there is also a magnetic force. This is because the moving charge q1 creates a magnetic field in the z direction which exerts a force on q2. This is certainly anomalous and is in apparent violation of the third law. We say it is an apparent violation because the third law is essentially a statement of conservation of momentum and it is the total momentum of the system that needs to be conserved. In this case, in addition to the two charges, there exists the electromagnetic field and if the field itself carries momentum, there is no violation. This is intuitive but the fact that electromagnetic field carries momentum is a fact.

Let us look at the force exerted on a system of sources (charges and currents) and electromagnetic field. Let  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev represent the moment associated with the sources. The force exerted on the system of sources is then given by Lorentz force equation

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Once again, for simplicity, we will consider linear electric and linear magnetic material. We will use Maxwell’s equations to cast these equations in in terms of field variables. We replace ρ sing Gauss’s law and the current density from Ampere’s law,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

and

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Substituting these in the force equation, we get,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Let us simplify some of the terms,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

In the last expression we have used Faraday’s law. Thus we have,

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

In the second term we have changed the order of cross product and hence a minus sign.

The last term on the right can be seen to be  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev where  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev is the Poynting vector. Since the expression has the dimension of force,  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev has the dimension of momentum density and we will identify this term as the rate of change of momentum associated with the electromagnetic field (radiation field) and take it to the left hand side to be added to the rate of change of momentum of the sources Prad.

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

The expression on the right looks asymmetric in the electric and magnetic quantities which can be rectified easily. Let us look at the electric field terms represented by the first and the third terms on the right.

The integrand is  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev other than for a factor of ∈In order to simplify this to desired form, we will calculate the Cartesian components of this. Let us find what its x component is and then we will add up the three components.

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

                                       Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

From symmetry, one can write the y and z components

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

Let us add the three components. The second term of each of the expressions add to give us

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

The remaining terms look messy and we will return to them shortly.

Let us consider the magnetic field terms. In order to make it symmetrical with the magnetic field, we need to add a term  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev to the triple vector product term. This is simply adding zero because  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev However, this will make the electric and the magnetic fields at par and we would get a contribution of  Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev similar to the case of the electric field. This would give us a term

Conservation of Energy and Momentum (Part - 1) Electrical Engineering (EE) Notes | EduRev

where u is the energy density of electromagnetic field. Thus this term (i.e. the gradient of the energy density) represents the momentum density term.

We will return to a discussion of the remaining terms in the next lecture.

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