Intensity and energy density, help ?

Question

An electromagnetic wave in a vacuum has a maximum magnetic field amplitude of 5.67×10−7 T. (a) What is the maximum electric field amplitude of the wave (V/m)? (b) What is the average power per unit area associated with the wave (W/m2)?

Answer 1

(a)

In an electromagnetic plane wave, electric field vector <E> and magnetic field vector <B> are always perpendicular to each other and the direction of propagation. Moreover, their amplitudes are related according to

B₀ = E₀ / c

(c speed of light)

Hence:

E₀ = c · B₀

= 3.0×10⁸ m/s · 5.67×10⁻⁷ T

= 170 V/m

(b)

The energy flux (W/m²) of an electro-magnetic field is given Poynting vector, which is defined as cross product of electric field vector and magnetic field vector:

<S> = (1/µ₀) <E> × <B>

(µ₀ magnetic constant)

In electromagnetic plane wave  <E> and <B> are perpendicular to each other. So <S> points in direction of motion and has the magnitude

S = (1/µ₀)·E·B

E and B follow a similar wave equation. The field strength at

position <r> are:

E(t,<r>) = E₀·cos(ω·t - <k>·<r>)  

B(t,<r>) = B₀·cos(ω·t - <k>·<r>)

=>

S(t,<r>) = (1/µ₀)·E₀·B₀·cos²(ω·t - <k>·<r>)  

Because the time- or space average of cos²(ω·t - <k>·<r>)

is ½, the average magnitude of the Poynting vector is given by:

S_av = (1/(2·µ₀))·E₀·B₀

Because E₀/B₀ = c

S_av = E₀² / (2·c·µ₀) = c·B₀² / (2·µ₀)

For this wave

S_av = (170V/m)² / (2 · 3.0×10⁸ m/s · 4·π×10⁻⁷ TmA⁻ ¹)

= 0.225W/m²