Displacement Current and Maxwell's Equations

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Chapter 17: Displacement Current and Maxwell’s Equations

Chapter Learning Objectives: After completing this chapter the student will be able to:  Use the continuity equation to determine the charge density or current density at a point.  List and explain the significance of Maxwell’s Equations.  Use Maxwell’s Equations to model the behavior of a magnetic field inside a conductor.

You can watch the video associated with this chapter at the following link:

Historical Perspective: James Clerk Maxwell (1831-1879) was a Scottish scientist who formulated a comprehensive mathematical model that unites electricity, magnetism, and light into one phenomenon. He is often considered the third greatest physicist of all time, after Newton and Einstein.

Photo credit: https://upload.wikimedia.org/wikipedia/commons/5/57/James_Clerk_Maxwell.png, [Public domain], via Wikimedia Commons.

17.1 The Continuity Equation

We have only one more piece of the puzzle to add before Maxwell’s Equations are complete. As you may recall, I have mentioned that we still need to add one more term to Ampere’s Law before it is in its final form. To understand the need for this additional term, let’s first consider the continuity equation.

The continuity equation can be derived from the conservation of matter. We know that if a net flow of particles is leaving a region, then this will necessarily cause a decrease in the density of those particles within the region. Stated another way, every particle that crosses the boundary of the region will (1) contribute to a positive flow out of the region, and (2) decrease the total population within the region. This principle is illustrated in Figure 17.1.

Figure 17.1. Particles leaving a region reduce the particles contained within the region.

When the “particles” are electrical charges, we can write an equation to model this relationship:

(Equation 17.1)

The negative sign in this equation represents the fact that positive current leaving a region will lead to a decrease in charge within the region. Equation 17.1 can be re-written in terms of a surface integral over current density and a volume integral over charge density:

(Equation 17.2)

The divergence theorem, which is true for any vector field, states that the surface integral of the vector field over a closed surface (the amount of the vector leaving the field) is equal to the volume integral of the divergence of the vector field (the net source of the vector field within the same region):

(Equation 17.3)

Applying this equation to the left side of Equation 17.2, we find:

(Equation 17.4)

Now, since both sides of this equation are volume integrals over the same volume, the functions being integrated must be equal:

(Equation 17.5)

This important result, known as the continuity equation, is typically written with both terms on the left side:

(Equation 17.6)

Since this result was derived from first principles (rather than from experimental observations), we know that it must be valid in all circumstances. If we find a case that violates it, then we will have to correct this problem.

Example 17.1: Assume that free charges are somehow injected into the interior of a conductor. We know that free charges in a conductor will move to the surface, leaving the interior empty of free charges and with zero internal electric field. Use the continuity equation to derive a function that describes how quickly this will happen. How long will it take the charge to dissipate to 1% of its initial value in copper ( = 6 x 107 S/m)?

17.2 Completing Ampere’s Law: Displacement Current

Recall that we have previously derived a point form of Ampere’s Law, which gives a relationship between current density J and magnetic flux density B.

(Copy of Equation 11.12)

However, this equation is incomplete. We can observe two problems with this form of the equation.

Problem #1: Disagreement with the Continuity Equation: If we take the divergence of both sides, we find:

(Equation 17.7)

But mathematicians have told us that the divergence of the curl of any vector field is always equal to zero, so this reduces to:

(Equation 17.8)

Yet, this disagrees with the continuity equation (Equation 17.6). Since we know that the continuity equation is always correct, there must be a problem with Equation 11.12.

Problem #2: Magnetic Field Near a Capacitor Consider Figure 17.2, in which an AC voltage source is connected to a capacitor. (I have drawn the capacitor to be extra-wide here to better illustrate the point.) We know that as current flows around the circuit, illustrated by the arrows, there will be a non-zero value of I and therefore of J, and Equation 11.12 tells us that there will be a magnetic field circulating around the wire at every point. I I=0 I B B=0? B B I I B

I B B I

B ~ B

I I V Figure 17.2. Is there a magnetic field between the plates of a capacitor?

But what happens between the plates of the capacitor? There is no actual current flowing between the plates, so from Equation 11.12, it would seem that there is an abrupt gap in the magnetic field, ending at one capacitor plate and beginning again at the other.

But that doesn’t match our experimental results. When we set up this circuit and measure the magnetic field, we find it circulating around every point of the wire (as expected), but we also find it circulating around the space between the capacitor plates (as definitely not expected). Where is this magnetic field coming from? The only thing between the plates is an electric field E (or D, if there is a dielectric), as shown in Figure 17.3.

E I I B B B

I I B B

B I I B

B ~ B I I V Figure 17.3. Why is there a magnetic field between the plates of a capacitor?

It turns out that changes to the electric field between the plates are causing the magnetic field. We can see how this comes to be by going back to the continuity equation:

(Copy of Equation 17.6)

We will then go way back to chapter 8, where we saw the final form of Gauss’s Law, including the effect of dielectric materials:

(Copy of Equation 8.6)

Substituting Equation 8.6 in the right side of Equation 17.6, we obtain:

(Equation 17.9)

We can then factor the divergence out of these two terms:

(Equation 17.10)

Notice how, in this equation, the partial derivative of D with respect to t seems to be a term very similar to the current J. This suggests that this derivative term might be treated in some ways as a current. We will call this term the displacement current, since it is related to the displacement field D:

(Equation 17.11)

With this new terminology, Equation 17.10 can now be rewritten as:

(Equation 17.12)

Remember that inside a conductor, E=D=0, so Jd will also be zero inside a conductor. If there is no electric field, there can be no displacement current. Conversely, inside an insulator or in free space, v=0, so conduction current J (the “normal” current caused by moving charges) is also equal to zero. Conduction current J only exists in conductors, and displacement current Jd only exists in insulators or free space.

We can now go back to the incomplete form of Ampere’s Law:

(Copy of Equation 11.12)

Notice how the right side includes a term for conduction current J but not for displacement current Jd. Maxwell’s greatest insight, which took him decades to discover, was to modify Ampere’s Law to include displacement current (which is the partial derivative of D with respect to t). Moving the permeability term to the left side and adding the displacement in derivative form, we get:

(Equation 17.13)

This is the final form of Ampere’s Law in Point Form, and it is the fourth and final of Maxwell’s Equations.

If we take the surface integral of both sides of Equation 17.13, we find:

(Equation 17.14)

Applying Stoke’s Theorem from the previous chapter to the left side, this becomes:

(Equation 17.15)

Then applying B=0H to the left side:

(Equation 17.16)

This is the final form of Ampere’s Law in Integral Form.

Example 17.2: Demonstrate that the magnetic field surrounding the capacitor in Figure 17.3 has the same magnitude as the magnetic field surrounding the conductor. Assume that

V(t)=V0sint.

17.3 Maxwell’s Equations Assembled

Maxwell really only added one piece to one of the four fundamental equations, but it really was the hardest piece, and it also happened to be the last piece, so he gets his name attached to the collection of four equations as a result. Here are the integral forms of Maxwell’s Equations:

(Faraday’s Law of Induction)

(Ampere’s Law with Displacement Current)

(Gauss’s Law)

(Gauss’s Law for Magnetic Fields i.e., no magnetic monopoles)

Here are the equations in point form:

(Faraday’s Law of Induction)

(Ampere’s Law with Displacement Current)

(Gauss’s Law)

(Gauss’s Law for Magnetic Fields i.e., no magnetic monopoles)

There are also three “helper” equations, called the Constitutive Equations:

(Effect of Dielectric Material)

(Effect of Ferromagnetic Material)

(Ohm’s Law)

There is one other equation that you should be prepared to use, which is the Lorentz Force Equation, describing the forces caused by both electric and magnetic fields:

Everything until this point has been building toward these 12 equations, and everything else from here on will be derived from one of these equations.

17.4 Applying Maxwell’s Equations to Determine Skin Depth

We can use Maxwell’s Equations to derive equations for many circumstances. In the next chapter, we will focus on what happens in free space, where the conduction current is equal to zero. This will give probably the most important result in all of electromagnetic field theory. But as a first example of the applications of Maxwell’s Equations, let’s consider what happens inside of a conductor like copper.

We will begin with Ampere’s Law. Since the displacement current is zero inside a conductor, this reduces to:

(Equation 17.17)

We will now substitute Ohm’s Law in the right side:

(Equation 17.18)

Taking the curl of both sides of this equation yields:

(Equation 17.19)

Substituting Faraday’s Law in the right side and converting H to B on the left side, we obtain:

(Equation 17.20)

Applying Equation 12.4 to this result gives:

(Equation 17.21)

Choosing the Coulomb Gauge, the first term on the left side goes to zero:

(Equation 17.22)

Rearranging constants gives:

(Equation 17.23)

Let us now consider the interface between a conductor and free space, as shown in Figure 17.4.

Bx(z) Free Space

Conductor az

a x Figure 17.4. Applying a sinusoidal magnetic field at the surface of a conductor

A sinusoidal magnetic field is being applied at the surface of the conductor. We know that this magnetic field will decay to zero inside of the conductor, but how far will it penetrate before it approaches zero? What happens to the magnetic field inside the conductor?

Since this problem really only has one dimension of variability (the z dimension), we can reduce the vector differential equation to a scalar one:

(Equation 17.24)

Since the magnetic field is sinusoidal, we can apply phasor methods to replace the time derivative on the right side with jBx(z):

(Equation 17.25)

2 If we then replace the constants 0j by  , we obtain a differential equation with a well-known solution form:

(Equation 17.26)

Which yields:

(Equation 17.27)

All we need to do now is solve for .

(Equation 17.28)

Because the right side of this equation is imaginary (from the j factor), g will have both a real and a complex component:

(Equation 17.29)

The  factor will control how quickly the wave decays, while the  factor will control how quickly it oscillates while decaying. In this case,  and  will have the same value.

(Equation 17.30)

The final solution will be an exponentially decaying sinusoid:

(Equation 17.31)

The wave will have decayed to 37% of its initial strength by the time it reaches 1/a, which is referred to as its “skin depth.”

(Equation 17.32)

6 Example 17.3: Sketch the wave that results when a magnetic field Bx(0)=10cos(10 t) is applied to the surface of a solid piece of copper ( = 6 x 107 S/m). What is the skin depth for this wave in copper?

Example 17.4: A microwave oven (f=2.45GHz) is sending waves into a sphere of water (30mS/m). At what depth will the wave strength have decayed by 50%?

17.5 Summary

 The continuity equation is derived from conservation of mass:

 Displacement current behaves very much like regular (conduction) current in creating magnetic fields:

 Including displacement current, we obtain the final form of Ampere’s Law in both point and integral form:

 This completes Maxwell’s Equations, which can be written in point or integral form:

 These equations are accompanied by three constitutive equations and the Lorentz force equation to provide a complete description of electromagnetic field theory:

 When applied to conductors, Maxwell’s Equations can be used to model the exponential decay of a sinusoidal magnetic field applied to the surface.