Sunlight warms your skin: electromagnetic waves carry energy. Each field holds energy, $\tfrac{1}{2}\varepsilon_0E^2$ for the electric field and $B^2/2\mu_0$ for the magnetic one, and the wave carries both along at $c$. Less obviously, light also carries momentum, so it pushes on whatever absorbs or reflects it.
Key idea
Average intensity (power per area):
$$I = \tfrac{1}{2}c\varepsilon_0E_0^2 = \frac{E_0B_0}{2\mu_0},$$
flowing along the Poynting vector $\vec{S} = \dfrac{1}{\mu_0}\vec{E}\times\vec{B}$. Light carries momentum $p = U/c$, so it presses on a surface with pressure $I/c$ if absorbed and $2I/c$ if reflected straight back.
Why it works
With $B = E/c$ and $c^2 = 1/(\mu_0\varepsilon_0)$, the magnetic energy density is $\dfrac{B^2}{2\mu_0} = \dfrac{E^2}{2\mu_0c^2} = \tfrac{1}{2}\varepsilon_0E^2$: exactly equal to the electric one. So the total is $u = \varepsilon_0E^2$, which averages to $\tfrac{1}{2}\varepsilon_0E_0^2$ over a cycle. This energy moves at $c$, so the energy crossing a square meter each second is $I = c\bar{u}$. For momentum: as light is absorbed, its electric field drives charges in the surface sideways, and the magnetic force $q\vec{v}\times\vec{B}$ on those moving charges points along the direction of travel. Working it through gives a push of $U/c$ for energy $U$ absorbed.
Worked examples
Worked example: Sunlight's fields
Sunlight above Earth's atmosphere delivers about $1360\ \text{W/m}^2$ (NASA). Treating it as one wave, find $E_0$ and $B_0$.
- $E_0 = \sqrt{\dfrac{2I}{c\varepsilon_0}} = \sqrt{\dfrac{2(1360)}{(3.00 \times 10^{8})(8.85 \times 10^{-12})}} = 1010\ \text{V/m}$.
- $B_0 = E_0/c = 3.37\ \text{μT}$, much less than Earth's own magnetic field.
Answer: $E_0 \approx 1010\ \text{V/m}$, $B_0 \approx 3.37\ \text{μT}$
Worked example: Pressure and a solar sail
Find the radiation pressure of that sunlight on an absorbing surface and on a mirror, and the force on an $800\ \text{m}^2$ reflecting sail.
- Absorbed: $I/c = 1360/(3.00 \times 10^{8}) = 4.53\ \text{μPa}$. Reflected: twice that, $9.07\ \text{μPa}$.
- Sail: $F = (9.07 \times 10^{-6})(800) = 7.25\ \text{mN}$. Tiny, but it never stops, and in space nothing slows the sail down.
Answer: $4.53\ \text{μPa}$, $9.07\ \text{μPa}$; $7.25\ \text{mN}$
Worked example: Spreading from a source
A lamp radiates $100\ \text{W}$ of light evenly in all directions. Find the intensity and $E_0$ at $2.0\ \text{m}$.
- The power spreads over a sphere: $I = \dfrac{100}{4\pi(2.0)^2} = 1.99\ \text{W/m²}$.
- $E_0 = \sqrt{2I/(c\varepsilon_0)} = 38.7\ \text{V/m}$. Intensity falls as $1/r^2$, so $E_0$ falls as $1/r$.
Answer: $I \approx 1.99\ \text{W/m²}$, $E_0 \approx 38.7\ \text{V/m}$
Common mistake
"Absorbed and reflected light push equally hard." The push equals the light's change in momentum. Absorbed light arrives with momentum $p$ and ends with none: change $p$. Reflected light leaves with $-p$: change $2p$. So a mirror feels twice the force, like a ball that bounces off a wall compared with one that sticks. Check the formulas: $I/c$ for absorption, $2I/c$ for reflection.