Maxwell's Equations
and Light

James Clerk Maxwell

• Born in Scotland (1831) and educated in Edinburgh and Cambridge.
• Went through an Evangelical conversion (1853).
• Argued (1859) mathematically that Saturn's rings must consist of small chunks.
• Laid off from his position as department chair in 1860 in the merger which resulted in the University of Aberdeen.

Magnetic field around a wire

Ampere's Law (1826)

`int_C`B`{::}_{||} d ell= mu_0 I_{enc} `     For an ~infinite wire: `rArr B = \frac{mu_0 I}{2\pi r}`

Is there a problem here?

If there is current in the wire, then either `B=(mu_0 I)/(2 pi r)` or `B=0` depending on the choice of surface. ... But is there any current at all?

Circuits with Capacitors

Current is moving charge. When the switch 'S' is closed...

• does current flow? In just one circuit or both?
• If current flows...does it ever stop?
• What does E`(t)` between the plates look like?

Maxwell notices a problem (1861)

We can arrange to have current in the wire. Then `B=(mu_0 I)/(2 pi r)` or `B=0` depending on the choice of surface.

Maxwell's proposed resolution

If `I` is the current flowing into (and out of) the capacitor, and we draw our surface `S` in between the plates where certainly `I=0`, we'd like something that...

Modified Ampere's law

`int_C``B`{::}_{||} d ell= mu_0 I + mu_0 epsilon_0 frac{d}{dt}int_S E_{_|_} da `

Maxwell's thinks on these four...

What if we combine equations to get rid of, say B, and find a new equation for only E?

E-field of an oscillating dipole

Close to the x-axis shown, and far from the dipoles, the electric field has only a `y`-component.

E`{::}_(-) +`E`{::}_+ = E_y(x,t) hat(y)`

Apply Faraday's law to small contour

`int_C`E`{::}_{||} d ell`	`=`	`-frac(d)(dt)int_S B_{_|_} dA`
`E_y(x_2) Delta y - E_y(x_1) Delta y	`=`	`-frac(d)(dt)B_z Delta x Delta y`
`(E_y(x_2) - E_y(x_1)) Delta y approx (frac(del E_y)(del x) Delta x) Delta y	`=`	`-frac(del B_z)(del t) Delta x Delta y`
`frac(del E_y)(del x)`	`=`	`-frac(del B_z)(del t)` ..BTW is there a `B_y` or `B_x`?

Apply Ampere's law to small contour

`int_C`B`{::}_{||} d ell`	`=`	`mu_0 epsilon_0 frac(d)(dt)int_S E_{_|_} dA`
`rArr frac(del B_z)(del x)`	`=`	`-mu_0 epsilon_0 frac(del E_y)(del t)`
`frac(del)(del t) (frac(del B_z)(del x))`	`=`	`frac(del)(del t) (-mu_0 epsilon_0 frac(del E_y)(del t))`
`frac(del)(del x) (frac(del B_z)(del t)) = frac(del)(del x)(-frac(del)(del x) E_y)= -frac(del^2)(del x^2)E_y`	`=`	`-mu_0 epsilon_0 frac(del^2 E_y)(del t^2)`

Wave equations

Springs:
`F=ma` for a spring is `k h =mfrac(d^2 h)(dt^2)`. Solution: `h(t)= h_0 sin 2pitf`

Sound:
`frac(del^2)(del x^2)P(x,t) = frac(1)(v^2)frac(del^2)(del t^2)P(x,t)`. Solution: `P(x,t) = P_0 sin 2pi(x/v -t)f`

This is also a wave equation:
`frac(del^2)(del x^2)E_y=mu_0 epsilon_0 frac(del^2 E_y)(del t^2)`

Electro-magnetic wave equation

`frac(del^2)(del x^2)E_y = mu_0 epsilon_0 frac(del^2 E_y)(del t^2)` is a wave equation!

The velocity is `frac(1)(sqrt(epsilon_0 mu_0)) approx 3 times 10^8`m/sec, pretty close to `c`, the speed of light.

Just a coincidence? Maxwell (1862): "We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena."

Unification

Maxwell connects electricity and magnetism with light.

His equations top the charts of greatest equations.

E-M spectrum

Problem: What's the B field?

Q: That solution for travelling sound waves suggests that a solution to the wave equation for E in the situation we've been looking at is:
E`=E_y(x,t) hat(y) = E_0 sin 2pi (x/c -t)f hat(y)`
Guess that the magnetic field is also a sinusoidal function like
B=`B_z(x,t) hat(z) = B_0 sin 2pi (x/c -t)f hat(z)`, and if that's the case...

What's the amplitude `B_0` in terms of `E_0`?

With...

`B_z = B_0 sin 2pi (x/c -t)f`, and `E_y = E_0 sin 2pi (x/c -t)f`

Use that relationship:
`frac(del B_z)(del x)= -mu_0 epsilon_0 frac(del E_y)(del t) = -1/c^2 frac(del E_y)(del t) ` to eventually find that `B_0 = E_0 / c`

Energy in E-M waves

E-M waves arise because of accelerated charges.

• Why do hot things glow? Why do hotter things glow with a higher frequency (shorter wavelength)?
• Can Electro-magnetic waves really propagate through a vacuum? What's the evidence?
• Do Electro-magnetic waves carry energy? What's the evidence?