Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) PDF Download

Coulomb Gauge and the Potential formulation of Maxwell’s equations

The Maxwell’s equations in their final form are

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Now the term  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is a current and we replace it by  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) using the equation of continuity and the term becomes

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) whose divergence is zero and another part  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) whose curl is zero,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Using the identity,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

we can write,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

where we have used  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) However, since this depends only on the difference of  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) we can replace Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) by Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)'by incorporating a minus sign. With this (I) becomes

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

We will simplify this further by using chain rule differentiation,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

With this (I) takes the form

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

As our fields vanish at infinity, we have,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Using this, we get,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

and

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Substituting these in the force equation, we get,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Let us simplify some of the terms,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is the Poynting vector. Since the expression has the dimension of force,  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

                                       Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

From symmetry, one can write the y and z components

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) to the triple vector product term. This is simply adding zero because  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) However, this will make the electric and the magnetic fields at par and we would get a contribution of  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) similar to the case of the electric field. This would give us a term

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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.

In the last lecture, we had written down the expression for the rate of change of momentum of the sources and fields as given by,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

We could express this equation in a symmetric fashion by adding a term prooportional to the divergence of the magnetic field (which is identically zero) and rewrite the above as

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

We considered the pairs of term in parenthesis on the right and simplified their components as

 Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Adding the second term of each of the components, we wrote it as  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and a similar contribution from the magnetic field gives us  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Inserting the associated constants ∈0 and  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) we get this contribution as the gradient of energy density Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)  which is the momentum density term.

We will now try to write the remaining terms in a compact fashion. We will simplify only the electric field terms and write down the corresponding magnetic field term by analogy. The remaining term are

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Since the left hand side is a force, we would have liked to express it as a gradient of a scalar which would act like a potential. However, looking at the terms does not seem to suggest such a possibility because the components seem to have been mixed.

We , therefore, bring back the terms which we have been able to express as a gradient of the energy density and see whether there can be alternative ways of expressing them.

We note here that in expressing it as a gradient of a scalar, we started with a single function and obtained three quantities, the components of a vector. we are aware that if we start with a vector and take a divergence, we would get a scalar. Thus in some sense, the gradient increases the space but a divergence decreases the same. We generalize the definition of a vector to a “Tensor” of arbitrary rank n as a collection of 3n quantities. Thus a scalar is a tensor of rank zero, a vector a tensor of rank 1. In this case, we take a tensor of rank 2, denoted by nine components  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) A divergence of a tensor of rank 2 will reduce its rank by 1, giving us a vector which has 3 components. 

We will first state the result and then show that this can be actually done. We claim that the right hand side of our force equation can be expressed as a divergence of a tensor of rank 2 denoted by  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) whose components are given by

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

To fix our ideas, let us once again, take B=0. We can express Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) as a 3 x 3 matrix as follows:

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Since the divergence of the tensor is a vector, it has three components and we define the components as follows

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Substituting the components given in the matrix above, we find,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

which is precisely the x component of the electric field term that we had found. Similar statement can be proved for the y and z components and also for the magnetic field terms. Thus we have,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

where, we have written da for the area element instead of dA so as not to confuse with the Poynting vector which we have denoted by  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The last relation is obtained by a procedure similar to the divergence theorem.

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is the momentum flux nornmal to the bounding surface of the volume. Since the relation is true for arbitrary volume, the relation can be stated in a differential form in terms of the momentum densities, indicated by  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) as

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

This is a statement of the conservation of linear momentum for the electromagnetic field which sates that the rate of change of momentum of a closed system containing sources and fields can occur only through a transport of momentum through the bounding surface.

Examples 

Force on one of the plates of a parallel plate capacitor

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Consider a parallel plate capacitor with charge densities  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Let the plates be in the yz plane so that the electric field is directed along the x-direction. we have,  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus the Stress tensor in this case is diaginal and can be written as 

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The force acting on the negative plate is (assuming it is to the right of the positive plate) is along the negative x direction as Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is directed along – x direction. We have,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Force on the Northern Hemisphere of a spinning charged sphere by the Souther Hemisphere 

Though solving this problem can be done by more straightforward method, we are doing it by the method of Stress Tensor as an illustration.

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

First, let us consider the spinning shell. As the angular velocity is constant, the spinning shell is equivalent to a magnetic moment directed along the direction of the angular velocity vector. Let us take the angular velocity vector along the z direction.

The moving charges on the surface are equivalent to circulating current in the  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) direction. The surface current density is given by, in spherical polar,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

where r=R  on the surface of the sphere. Consider a thin circular strip lying between angles θ and  θ + d θ.The radius of the circle is R sin θ.The width of the strip (which is crossed by the linear current density) is Rd θ so that the current flowing in the strip is  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The circulating loop is equivalent to a magnetic moment (πRsin2θ)dl in the z- direction. Since all such loops are parallel, the net magnetic moment of the spinning shell is

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The magnetic field due to the shell is given by

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Since the fields are constant, so is the Poynting vector. Thus the force is given by

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

By symmetry, the force is in z direction. Thus we will need to calculate only the z component of the normal component of the stress tensor. In this case, we consider only the magnetic forces (the electric force is also there as the shell has charges, but we restrict ourselves to magnetic case only. Since we are interested only in the z components, the only components of the stress tensor that we requires are

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

In terms of these, we can write down the force as

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

As we are interested in calculating force on the northern hemisphere, there are two surfaces to take into account. The equatorial plane inside the sphere for which the normal direction is along the negative z direction. Since we need to consider only the inside field, the field to be considered here is constant.

The field is given by  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus only term in stress tensor is  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and as the field is constant, the force is

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

That leaves us with the outside hemisphere. The normal to the surface is along the radial direction. We need to compute the integral  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Let us compute each term of the integrand using the expression for the field outside the shell, which is given by

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Since  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) we have,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Using these,

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)   

the limits of integration are from 0 to π/2 as it is a hemisphere.

The second integral is  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and is given by, using  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Adding the two contributions, the force on the outer surface is

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Substituting the value of the magnetic moment,  Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

If we add both the inside and the outside contributions, the magnetic force on the northern hemisphere is

Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

the negative sign shows that the force is directed towards the southern hemisphere and is attractive in nature.

The document Conservation of Energy & Momentum | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Electromagnetic Fields Theory (EMFT).
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FAQs on Conservation of Energy & Momentum - Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

1. What is the law of conservation of energy?
Ans. The law of conservation of energy states that energy cannot be created or destroyed, but it can only be transferred or transformed from one form to another. This means that the total amount of energy in a closed system remains constant over time.
2. What is momentum and how is it conserved?
Ans. Momentum is a property of moving objects and is calculated as the product of an object's mass and velocity. The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces are acting on it. This means that the total momentum before an event is equal to the total momentum after the event.
3. How does the conservation of energy relate to the conservation of momentum?
Ans. The conservation of energy and the conservation of momentum are both fundamental principles in physics. While the conservation of energy focuses on the overall energy of a system, the conservation of momentum focuses on the overall momentum of a system. Both principles help us understand and predict the behavior of objects and systems.
4. Can energy be converted into momentum or vice versa?
Ans. Yes, energy can be converted into momentum and vice versa. For example, when an object gains kinetic energy, its momentum also increases. Similarly, an object with momentum can transfer that momentum to another object, causing it to gain energy. The conversion between energy and momentum is governed by the laws of physics.
5. What are some real-life examples of conservation of energy and momentum?
Ans. There are several real-life examples of the conservation of energy and momentum. One example is a collision between two billiard balls. When the balls collide, their total momentum before the collision is equal to their total momentum after the collision, assuming no external forces are involved. Another example is a roller coaster ride, where the potential energy at the top of a hill is converted into kinetic energy as the coaster descends. This conversion of energy and the conservation of momentum ensure a thrilling and safe ride.
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