Intensity of Light

Consider a plane, linearly polarized electromagnetic wave. Using complex representation, the fields in the wave are given by

$$\vec{E}(x,t) = \Re (\underline{E}(x)\,e^{-i\omega t})\,\bf {\hat{y}},$$

$$\vec{B}(x,t) = \frac{1}{c} \Re (\underline{E}(x)\,e^{-i\omega t})\,\bf {\hat{z}},$$

where $\underline{E}(x)$ is the complex amplitude of the wave at point $x$.

(a)

Compute the densities of electric field energy and magnetic field energy at $x$ as a function of time $t$. Simplify your answers so that they do not contain any complex numbers.
HINT: Write the complex amplitude $\underline{E}(x)$ in the polar form and use Euler's formula.

(b)

Compute the total energy density at $x$.

(c)

Compute the power flux vector (usually called ``Poynting vector'' in eletromagnetic theory) $\vec{S}(x,t) \equiv \frac{1}{\mu} \vec E \times \vec B$. Which direction is energy flowing in? Express the magnitude of the Poynting vector, $S(x,t)\equiv\vert\vec{S}(x,t)\vert$, in a form that does not contain any complex numbers.

(d)

Show that $S(x,t)$ is a periodic function of time. How is the period of this function related to the period $T$ of the electromagnetic wave itself ( $T=2\pi/\omega$)?

(e)

The wavelength of visible light lies in the range between 400 nm (blue) and 700 nm (red). Find the period $T$ of electric and magnetic fields in blue and red light waves.

(f)

Intensity $I(x)$ is obtained by averaging $S(x,t)$ over time. Using the results from (d) and (e), explain why $I(x)$ provides a better measure of the ``amount of light'' seen by a human eye than the quantity $S(x,t)$.

(h)

Using your result from part (d), obtain the formula that we have used to study interference,

$$I(x)=\alpha \vert\underline{E}(x)\vert^2.$$

What is the value of $\alpha$ in terms of $\epsilon_0$ and $\mu_0$?
HINT: Time-averaged values of $\cos^2 (\omega t+\phi_0)$ and $\sin^2 (\omega t+\phi_0)$ are both equal to $1/2$.