An electromagnetic wave in a vacuum has a magnetic field with a magnitude of 1.7 x 10^-6 T. What is the intensity S of the wave?
To find the intensity of the electromagnetic wave, we can use the relationship between intensity and the magnetic field strength.
The formula to calculate the intensity (S) of an electromagnetic wave is given by:
S = (c * μ₀ * B²) / 2
Where:
- S is the intensity of the wave
- c is the speed of light in a vacuum (approximately 3 x 10^8 m/s)
- μ₀ is the permeability of free space (approximately 4π x 10^-7 T·m/A)
- B is the magnitude of the magnetic field strength
Substituting the given values into the formula:
S = (3 x 10^8 m/s * 4π x 10^-7 T·m/A * (1.7 x 10^-6 T)²) / 2
Now, let's calculate:
S = (3 x 10^8 m/s * 4π x 10^-7 T·m/A * 2.89 x 10^-12 T²) / 2
First, multiply the numbers in the numerator:
S = (3 x 4π x 2.89 x 10^-7 x 10^-12) T²·m²/A
Simplifying further:
S = (8.748 x 10^-19) T²·m²/A
Since the unit of intensity is Watts per square meter (W/m²), we can rewrite the final answer as:
S ≈ 8.75 x 10^-19 W/m²
Therefore, the intensity of the electromagnetic wave is approximately 8.75 x 10^-19 Watts per square meter (W/m²).
In plane waves
In a propagating sinusoidal linearly polarized electromagnetic plane wave of a fixed frequency, the Poynting vector always points in the direction of propagation while oscillating in magnitude. The time-averaged magnitude of the Poynting vector is
\langle S \rangle = \frac{1}{2 \mu_0 c} E_0^2 = \frac{\varepsilon_0 c}{2} E_0^2,
where \ E_0 is the maximum amplitude of the electric field and \ c is the speed of light in free space. This time-averaged value is also called the irradiance or intensity I.
[edit] Derivation
In an electromagnetic plane wave, \mathbf{E} and \mathbf{B} are always perpendicular to each other and the direction of propagation. Moreover, their amplitudes are related according to
B_0 = \frac{1}{c}E_0,
and their time and position dependences are
E\left(t,{\mathbf r}\right) = E_0\,\cos\left(\omega\,t- {\mathbf k} \cdot {\mathbf r} \right),
B\left(t,{\mathbf r}\right) = B_0\,\cos\left(\omega\,t- {\mathbf k} \cdot {\mathbf r} \right),
where \ \omega is the frequency of the wave and \mathbf{k} is wave vector. The time-dependent and position magnitude of the Poynting vector is then
S(t) = \frac{1}{\mu_0} E_0\,B_0\,\cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r}\right) = \frac{1}{\mu_0 c} E_0^2 \cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r} \right) = \varepsilon_0 c E_0^2 \cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r} \right).
In the last step, we used the equality \varepsilon_0\,\mu_0 = {c}^{-2} . Since the time- or space-average of \cos^2\left(\omega\,t-{\mathbf k} \cdot {\mathbf r}\right) is \textstyle\frac{1}{2}, it follows that
\left\langle S \right\rangle = \frac{\varepsilon_0 c}{2} E_0^2.
E= speedlight*B
The time averaged Poynting vector is equal to epsilonfreespace*speedlight*E^2/2
but if E= speedlight*B
S= epsilonFreespace*speedlight^3*B^2 /2
check this out: http://en.wikipedia.org/wiki/Poynting_vector