Chapter 34 – Electromagnetic Fields and Waves

This chapter ties together much of went before and condenses the behavior of electric and magnetic fields into a concise set of mathematical statements known as Maxwell’s equations. These equations encode everything covered in class and along with the force on a charged particle, F = q(E + vxB), can in principle describe all the classical interactions between a system of arbitrarily moving charges.

1. The chapter begins with a rather clever discussion of how observers moving with respect to one another measure “different” electric and magnetic fields. More specifically, Sharon is usually moving with velocity V with respect to Bill who is assumed for the most part to be stationary. For example, suppose Sharon is carrying a positively charged ball toward Bill, from left to right. Further suppose that Bill has created a magnetic field pointing into the page. Therefore when Sharon moves through Bill’s laboratory with the magnetic field, Bill sees an upward magnetic force thrust the ball out of Sharon’s hand. From Sharon’s perspective, the ball is stationary and consequently cannot feel a magnetic force. Therefore when the ball jumps out of her hand, she concludes that there must be an electric field acting on the stationary ball.

2. After reviewing a few examples like this, see the text, we can deduce the relationship between the fields at a particular spot as measured by Sharon and Bill:

E’ = E + VxB E = E’ - VxB

B’ = B - VxE/c² B = B’ + VxE/c²,

where c = 3 x 10⁸ m/s, the speed of light and ε₀ μ₀ = 1/c².

3. The above equations are true as long as v << c which is going to be true in most earthly situations because c is a very, very large speed! At speeds approaching the velocity of light, the relationship between what Sharon measures and what Bill measures now require special relativity which is introduced in chapter 36 but is not part of this course.

4. The primary point of the above discussion was to reinforce the idea that there is a fundamental linkage between magnetic and electric phenomena and also that the fields, abstract entities early in the semester, have substance and can exist independent of their sources. They are real things in the sense that they can be measured and can carry energy and momentum as seen below.

5. There are four Maxwell equations. The first two are “Gauss’ Law” for electric and magnetic charges. Since north magnetic poles are always associated with south magnetic poles, it is impossible to have an isolated magnetic charge. Consequently, when the flux of the magnetic field is calculated through a closed surface, the integral gives zero since there is no net magnetic charge inside the enclosed volume. Gauss’ Law for electric and magnetic fields can be summarized as:

∫E^. dA = q/ε₀ and ∫B^. dA = 0.

6. The next two of Maxwell’s equations concern the way a changing magnetic field creates an electric field, an equation that encapsulates Faraday’s and Lenz’ laws, and the way a changing electric field produces a magnetic field, a version of Ampere’s law modified by Maxwell. To be able to capture the content of Faraday and Lenz’ laws in a single equation we need to be able to unambiguously assign a direction to the surface used to calculate the magnetic flux ∫B^. dA . Note the difference between this surface integral and the one above in the statement of Gauss’ law for magnetic fields is that the previous calculation of the flux was through a closed surface. A closed surface has an unambiguous outward pointing normal. The surface integral in Faraday’s law is over an open surface, a circle for example. If the circle is lying in the plane of this page, a normal perpendicular to the surface can point into or out of the page. We need to resolve that ambiguity because if the magnetic field points into the page, using the normal pointing out of the page gives a negative flux while using the normal that points into the page gives a positive flux!

7. To figure out a sign convention let’s focus on a concrete example. A solenoid has its axis perpendicular to this page. Viewed along the axis, the current goes around the solenoid in a clockwise (cw) direction. The magnetic field produced by the solenoid points into the page. If the current is increasing, B is increasing into the page. Now imagine a loop outside the solenoid. The emf produced on the loop due to the changing magnetic flux through the loop opposes the increase in flux. Therefore the induced current that would be produced by that emf wants to generate a magnetic field pointing out of the page. The resulting current in our loop would be counter clockwise (ccw). The emf = ∫ E ^. ds, a line integral around a loop centered on the solenoid. Since the current is ccw, the electric field points in that sense since it is what drives the charges to produce the induced current. Integrate around that loop in the same sense as E, that is ccw, so that E and ds are parallel at every point on the circular loop so that E ^. ds > 0. We use the right-hand rule to define a direction for the dA used to calculate the flux. Grab the loop surrounding the solenoid with your right hand with your thumb point in the direction of ds, ccw in this example. Your fingers poke out of the page defining the direction of dA. The increasing magnetic field in the solenoid points into the page while dA points out of the page, so B ^. dA < 0. The only way the emf which is positive can equal the time rate of change of flux, which is negative is if the equation includes a minus sign,

∫ E ^. ds = - d/dt(∫B^. dA ).

This is the third of Maxwell’s four equations. The same sign convention will be used in the fourth and last equation that connects a magnetic field to currents and changing electric flux.

8. The last equation demonstrates some of Maxwell’s genius. When Maxwell thought about a circuit consisting of a battery, switch, resistor, and capacitor, he saw that during the short time the capacitor took to charge, remember the time constant for an RC circuit is τ = RC, a current flowed in the circuit. This current produced a magnetic field around the wires in the circuit, B = μ₀ I/2πr. This field surrounded the wire on each side of the capacitor but since no current flowed across the capacitor, there was no magnetic field inside the region bounded by the capacitor. The discontinuity of the magnetic field seemed illogical to Maxwell. Therefore he proposed a way to make the magnetic field more continuous by adding a term to this equation. The new term was due to the changing electric field in the capacitor. Maxwell said that changing electric flux in the capacitor produced a “displacement current” and that new kind of current produced a magnetic field equivalent to the one produced on the wires leading to the capacitor. With the addition of this term to Ampere’s Law, the revised equations predicted the existence of electromagnetic waves which were one of the greatest, if not the greatest, achievements of physics in the 19^th century.

9. To see what Maxwell did, we start with Ampere’s law, ∫B ^. ds = μ₀ I, the line integral of the magnetic field about a loop is equal to the net current passing through a surface bounded by that loop. To find the magnetic field due to a wire, choose the loop to be a circle centered at the wire and the surface is typically taken as the circular area inside the loop. But there is no reason that the surface has to lie in the plane of the loop. For example, we could use a hemisphere as the surface bounded by the circle. Imagine a wire perpendicular to the page with a current going into the page. The resulting magnetic field forms circles around the wire with the field direction being cw. The net current passing through the surface bounded by the circle is the same whether the surface is a circle in the plane of the page or a hemisphere with the circle as a base. Now imagine a wire leading to a capacitor. The loop is drawn around the wire where we know the value of ∫B ^. ds but the hemisphere is drawn so it passes between the plates of the capacitor where the current is zero. So we have the embarrassing situation where something, ∫B ^. ds, is equal to zero according to Ampere’s law. Maxwell solved this dilemma by adding a term to Ampere’s law that depends on the time rate of change of the electric flux, a term analogous to the flux term in Faraday’s law,

∫B ^. ds = μ₀ I + μ₀ε₀ d/dt(∫E^. dA ) = μ₀ I + μ₀ε₀ dΦ_E/dt.

10. Maxwell’s four equations are shown below:

∫E^. dA = q/ε₀

∫B^. dA = 0.

∫ E ^. ds = - d/dt(∫B^. dA )

∫B ^. ds = μ₀ I + μ₀ε₀ d/dt(∫E^. dA )

11. These equations predict the existence of plane electromagnetic waves traveling in the positive x-direction,

E(x, t) = E₀ sin[2π(x/λ - f t)] j

B(x, t) = B₀ sin[2π(x/λ - f t)] k

Notice that E and B are each functions of both x and t. Also λ = the wavelength and f = the frequency of the wave that travels with speed v_em= f λ. Also the electric and magnetic fields are perpendicular to one another and the direction of motion of the wave is given by E x B since j x k = i. In the book, it is shown that the above electric and magnetic fields are consistent with Gauss’ law for electric and magnetic fields while Faraday’s law requires that E₀ = v_em B₀ and the Ampere-Maxwell equation shows that v_em² = 1/μ₀ε₀ = c².

12. The rate at an electromagnetic wave carries energy is given by the Poynting vector,

S = E x B /μ₀ Watts/m².

For the plane wave above,

S(x, t) = (E₀ B₀ /μ₀){sin[2π(x/λ - f t)] }² i = (E₀²/μ₀c){sin[2π(x/λ - f t)] }² i

13. The frequency of an electromagnetic wave is so large that the usual quantity of interest is the time-averaged value of S. This is called the wave intensity I,

I = ½ (E₀²/μ₀c) because the average value of sine squared over a cycle is ½.

14. When light is absorbed it transmits momentum to the absorbing object, that momentum transfer produces a radiation pressure on the object,

Pressure due to Radiation = I/c = intensity/speed of light.