## Sunday, March 8, 2015

### Solutions to Electrodynamics (2)

Solutions (2):

1) We have:   B = E /c

=   (220 V/ m)  /  (3 x 10 8 m/s) =   7.3  x 10-7  T

2)      The intensity of an EM wave is :

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

where the area A = 2.4 m x 0.7 m = 1.68 m 2  and P
R  denotes the radiation pressure, or   P R = S/c (for complete absorption)

Now, S = Power/ area )=  ( 25 W/m 2 ) / (1.68 m 2) = 14.88 W

P
R = S(av) / c = 14.88 W/ (3 x 10 8 m/s)

= 3.3 x 10  - 9  N/m2

(3) We have: P(av) = 4 kw = 4 x 10 3 W

Assume spherical symmetry for area affected so area of the applicable spherical surface is:  4 π r 2

With r the distance (radius) to  the radio station transmitter but convert to m :

r  = 4 mi. =

4 mi (5280 ft./mi)(0.303 m./ft)  =    6.4 x 103   m

Area = 4 π ( 6.4 x 103  m) 2 = 5.1  x 10 8 m2

Therefore the average Poynting vector associated with the transmission is:

S(av) = P(av)/ A =

(4 x 10 3 w)/ (5.1  x 10 8 m ) =  7.7 x 10 -6  W/m2

But recall:

S(av) =  S(av) = E(max) 2/ 2
mo c

Therefore, solving for E(max):

E(max) = [2S(av)
mo c] ½

Then: E(max)=

[2(7.7 x 10 -6  W/m2 ) (4 π x 10 -7  H/m)(3 x 10 8 m/s)] ½

E(max) = 0.076   V/m

Then the emf induced in a 65 cm long (L =0.65m) antenna is:

Emf = E(max) L = (0.076 V/m) x (0.65m) = 0.049  V

(4) We assume: P(solar) = 10 3 W/m2

But because efficiency is relevant we need P(in). Thus,

eff = P (out)/ P(in) = 0.3 = 1 MW/ P(in)

Where 1 MW = 10 6 watts is the desired energy to come out, or be produced.

To get this, the power we need to put in, is:

P(in) = P(out)/ eff = (10 6 w)/ 0.3 = 3.33 x 10 6 W

Then:

Area A = P(in)/ S =

(3.33 x 10 6 W)/ (10 3 W/m2)    = 3.33 x  10 3  m2

Or about 3, 330 square meters.

5 ) By definition of current: I = nevA

where v is the drift velocity: v = I/neA

Where A = z1 y1 = (2 cm) (1 mm) = 0.02m (0.001 m) =

(2 x 10  -2 m) (1 x 10  -3 m) = 2 x 10  -5 m 2

Therefore:

v = (200A)/ [(7.4 x 10 28 /m )(1.6 x 10 -19 C)(2 x 10  -5 m 2)]

v = 8.4 x 10 -4 (-x^) m/s

b) E =
VH / z1

where z1= 0.02m (= 2 cm)

But:
VH = BI/ new

= (1.5T) (200 A) / [(7.4 x 10 28 /m )(1.6 x 10 -19 C)( 1 x 10  -3 m)]

= 2.5 x 10  -5   V (-z^)

Then: E =  VH / z1  =

2.5 x 10  -5  V (-z^)/ 0.02 m = 1.2 x 10  -3  V/m(-z^)