Units

We will use MKS units in this course. meter second kilogram volt ampere ohm electron-volt

Electrical units can be confusing. We will break down some of the units (farad, henry, tesla) and see what's inside them.

Maxwell's Equations in MKS units.

∇⋅D = 

∇⋅B = 0 Why this asymmetry to the first equation?

∇×E = −B˙

∇×H = J D˙

−7 D = 0 EP 0 = 4 ⋅10 Henries/meter B =  H M −12 0 0 = 8.85⋅10 Farads/meter

Force = qEv×B

 Z = 0 = 377 0  Impedance of free space. (What does this mean?)  0 1 c= 0 0

Let's Look at electrical units, with some memory aids Start with a simple equation that uses that quantity. coulomb amp⋅sec sec Q = C V [ farad ] = = = volt volt ohm volt⋅sec V = L I˙ [henry] = = ohm⋅sec amp volt⋅sec B˙ A=V [tesla] = m2 volt V =I R [ohm] = amp amp ∇×H = J [ H ] = m henries B volt⋅sec ohm⋅sec B =  H [ ] = = = = 0 0 meter H amp⋅m m 1 farads amp⋅sec  ⋅ = [ ] = = 0 0 c2 0 meter volt⋅m  volt⋅sec volt⋅m volt 2 Z = 0 [Z ] = ⋅ = = ohm 0  0 2  0 amp⋅m amp⋅sec  amp Ohm's Law: E = IR, Easy, I Remember. But E is reserved for electrical field strength, so we use V = IR (Verily, I Remember).

Carry units along in your calculations. It can help catch errors. A Practical Example

I came across the following equation the other day (frequency dependence of single-point multipactoring), where f is frequency, B is magnetic field and m is the mass of the electron.

f e B = 0 N 2  m

Instead of having to look up the physical constants, and knowing that the units of magnetic field B are volt-seconds/meter2, I recast the equation as:

2 2 f e c B0 e m volt−sec = N  mc2  2 [ e−volt sec2 m2 ]

2 2 The units of the (e/mc ) are 1/volt, and of c B0 are volt/second, and I know that the value of (e/mc2) is 1/(511000 volts), so the units come out okay (sec-1) and I don't have to look up the mass of the electron or its charge.

10 -19 f/N = 2.8x10 . for B0 = 1 Tesla. If you use e and m, e = 1.6x10 coul, -31 and me = 9.11x10 kg, and you get the same answer, but I had to look it up.

Wave Equation in Free Space

Let's determine the relationship between the electric and magnetic field vector in free space.

∇×E = −B˙ 1 ∂ E ∇×∇×E = −∂ ∇×B = ∂ t c2 ∂ t

1 For a plane wave in x Remember: ∇ ×H = D˙  ∇ ×B = E˙ c2

2 2 d E x 1 d E x = d z2 c2 d t 2

And a similar equation for B. These equations can be solved by

i kz−t  ˙ ¨ 2 E x = E 0 e E x = −i  E x E x = − E x i kz−t ˙ 2 B y = B0 e B y = −i  B y B¨ y = − B y

Plane Wave

 Define a wavenumber k, increases by 2p for each wave of length l. 2 1 f  k = , f =c , = =   c 2 c

We can rewrite the plane wave as

 i z − t  z t ikz−t  2i  −  1 E = E e = E e c = E e   = x 0 0 0 f

For a constant time t, moving along z gives one oscillation period per wavelength l.

Sitting at a particular location z, one oscillation occurs for every period t = t.

One can ride along a particular phase at velocity c (in the lab) as time progresses.

Ratio of E to H in a plane wave dE Back to Maxwell's equations B˙ = −∇×E = y dz

 i z − t dE  E = E e c x = i E x 0 dz c x  i z − t c ˙ B y = B0 e B y = −i  B y

 −i  B =i E y c x

B = 0 H

E =  c = Z H 0 0

The ratio of E to H fields in free space is Zo, the free-space impedance. The units are an indication: volts/meter / amps/meter = volts/amps = ohms