Wednesday, March 23, 2011

Introducing Basic Electrodynamics (2)

We left off in the last instalment with the two plane wave equations:

E= E(max) cos (kx - wt)

B = B(max) cos (kx - wt)

Taking partial derivatives in each case, the first with respect to x, the second with respect to t:

@E/@x= -k E(max) sin (kx - wt)

-@B/@t= -w B(max) sin (kx - wt)

Since: @E/@x = -@B/@t then:

-kE(max) = -w B(max) or, E(max)/B(max)= w/k

Also, w/k = (2 πf/ 2 πf/L) = c = E/B (where L is the wavelength of the EM wave)

that is, the ratio of the field maximum amplitudes is equal to the ratio of the angular frequency (w = 2 πf) to the wave number vector, k(2 πf/L ). Bear in mind the units of E are in V/m (volts per meter) and for B, Tesla.

Energy Carried by E-M Waves:

The energy carried by an electromagnetic (E-M) wave with field intensities E, B is given by the Poynting vector, S:

S = 1/u_o [E X B]

where again, u_o denotes the magnetic permeability, or u_o = 4 π x 10^-7 H/m.

Physically, the Poynting vector is the rate at which energy flows through a unit surface perpendicular to the flow.

Recall now, for any two vectors A, B

A X B = A B sin Θ

where Θ is the angle between them. If then A is perpendicular to B, then:

sin Θ = sin (90) = 1 so that:

A B sin Θ = AB

By the same token, taking the vectors E, B perpendicular to each other, we find:

S = EB/ u_o

which has units of watts/m^2

Now, recall from earlier, E/B = c, so:

B = E/c

Then: S = EB/ u_o = (E/c) E/u_o = E^2/ c u_o

or: S = (c/u_o) B

The "time average" is also of interest and entails taking the time average of the function: cos^2(kx - wt) (why?)

Which yields:

T(av) {cos^2(kx - wt) } = ½

The average value of S (or Intensity) can then be obtained from the maximum vector amplitudes, viz.

I = S(av) = E(max) B(max)/ 2 u_o


S(av) = E(max)^2/ 2 u_o(c) = cB(max)^2/ 2 u_o

Note that u_o c is a very important quantity known as the "impedance of free space", or:

u_o c = [u_o/e_o]^½

u_o c = [4 π x 10^-7 H/m/ 8.85 x 10^-12 F/m]^½ = 377 ohms

The respective contribution of the two field energies (associated with the electric intensity, E, and magnetic intensity B) can easily be shown to be:

U(E) = ½ e_oE^2

U(M)= ½ (B^2/ u_o)

In a given volume the energy is equally shared by the two fields such that:

U(E) = U(M) = ½ e_oE^2 = (B^2/2 u_o)

The total, instantaneous energy density of the fields is then:

U = U(E) + U(M) = 2(½ e_oE^2 ) = e_oE^2 = B^2/u_o

Averaged over one or more cylces this leads to the total average energy per unit volume:

U(av) = [e_oE^2]av = ½ e_oE(max)^2= B(max)^2/2 u_o

The intensity of an EM wave is then:

I = S (av) = c U(av) = PA

where P denotes the radiation pressure, or P = S/c (for complete absorption) or P = 2S/c for complete reflection of the wave.

(In direct sunlight, one finds P_R = 5 x 10^-6 N/m^2)

Some Problems:

1) An E-M wave in a vacuum has an electric field amplitude E = 220 V/m. Compute the magnitude of the corresponding magnetic field, B.


We have: B = E/c = 220 V/m/ (3 x 10^8 m/s) = 7.3 x 10^-7 T

2) An isotropic electromagnetic wave source has a power of 100 watts. At what distance from the wave will the maximum electric intensity E(max) be 15 V/m?

Recall: S(av) = E(max)^2/ 2 u_o(c)

But we define S(av) = Power/area, where in this case Power = 100w


Power = S(av) area

for an isotropic volume or space for which all distances are equal we use a sphere of radius r, and we know the area of a sphere is:

A= 4 π r^2


Power = S(av) 4 π r^2 = {E(max)^2/ 2 u_o(c)} 4 π r^2

We obviously need to solve for r, so use algebra to re-arrange and obtain:

r = [100w (u_o) c/ 2π E(max)^2 ]^½

substituting all values:

r = [100w (4 π x 10^-7 H/m) (3 x 10^8 m/s) / 2π (15 V/m)^2 ]^½

r = 5.1 m

3) If the amplitude of the magnetic field in an EM wave is B(max)= 4.1 x 10^-8T, then what is the average intensity of the wave?

We have: I = S (av) = c U(av)


U(av) = B(max)^2/2 u_o


I = S (av) = c B(max)^2/2 u_o

I - (3 x 10^8 m/s)(4.1 x 10^-8 T)^2 / 2(4 π x 10^-7 H/m) = 0.201 W/m^2

Problems for Energized readers:

1) A radio wave transmits 25 W/m^2 of power per uit area. A plane surface of area 2.4 m x 0.7 m is perpendicular to the direction of propagation of the wave. Calculate the radiation pressure P_R on the surface if it is assumed to be a perfect absorber.

2) An AM radio station broadcast isotropically with an average power of 4 kW. A dipole receiving antenna 65 cm long is located 4 miles from the transmitter. Find the Emf induced by this signal between the ends of the receiving antenna.

3) A community plans to build a solar power conversion station, i.e. to convert solar radiation into electrical power. They require 1 MW (megawatt) of power, and the final system is assumed to have an efficiency of 30% (30% of the solar energy incident on the surface is converted to electrical energy). What must be the effective area, A, of an assumed perfectly absorbing surface to be used in such an installation? Assume a constant solar energy flux incident of 1000 W/m^2.

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