Problem — Amplitudes of the solar wave

Problem

Sunlight at ground level has intensity $I\approx 1{,}0$ kW/m². (a) Derive the amplitude $E_0$ of the electric field from $I=\tfrac12\varepsilon_0 c E_0^2$. (b) Derive the amplitude $B_0=E_0/c$. (c) Compare $B_0$ with Earth’s magnetic field $\sim 5\cdot 10^{-5}$ T: which is stronger?

Solution

(a) Electric field amplitude. We isolate $E_0$ from the intensity formula:

$E_0=\sqrt{\frac{2I}{\varepsilon_0 c}}$

Substituting $I=1{,}0\cdot 10^3$ W/m², $\varepsilon_0=8{,}85\cdot 10^{-12}$ F/m, $c=3{,}00\cdot 10^8$ m/s:

$E_0=\sqrt{\frac{2\cdot 1{,}0\cdot 10^3}{8{,}85\cdot 10^{-12}\cdot 3{,}00\cdot 10^8}}\approx\sqrt{7{,}5\cdot 10^5}$

$\ev{E_0\approx 8{,}7\cdot 10^2\;\text{V/m}\approx 870\;\text{V/m}}$

(b) Magnetic field amplitude. In an EM wave in vacuum $B_0=E_0/c$:

$B_0=\frac{870}{3{,}00\cdot 10^8}\approx 2{,}9\cdot 10^{-6}\;\text{T}$

$\ev{B_0\approx 2{,}9\;\mu\text{T}}$

(c) Comparison with Earth’s field.

$B_0\approx 2{,}9\;\mu\text{T}<B_\text{Terra}\approx 50\;\mu\text{T}$

Earth’s magnetic field is about $17$ times stronger than the magnetic field of the sunlight wave.

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Topics: Onde elettromagnetiche Concepts: Intensità di un’onda · Campo magnetico · Campo elettrico