Work from the bench to a solar sail

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Maxwell's equations and conservation laws · Electromagnetic waves and their propagation

Electricity, Magnetism and Electromagnetism, Ch Plane electromagnetic waves: polarisation, energy flux and radiation pressure

Problem

Work from the bench to a solar sail. A parallel-plate capacitor with vacuum between its plates has plate area A=0.240 m2A = 0.240\,\mathrm{m^2} and gap d=1.20 mmd = 1.20\,\mathrm{mm}; its capacitance is measured as C=1.771 nFC = 1.771\,\mathrm{nF}, and edge effects are ignored so that C=ε0A/dC = \varepsilon_0 A/d. A separate measurement of the force between currents gives μ0=1.2566×10−6 H m−1\mu_0 = 1.2566\times10^{-6}\,\mathrm{H\,m^{-1}}. The solar constant at the Earth is 1.36 kW m−21.36\,\mathrm{kW\,m^{-2}}. ((a)) Obtain ε0\varepsilon_0 from the capacitor measurement. ((b)) For source-free fields depending only on zz and tt, Maxwell's equations give ∂Ex∂z=−∂By∂t,∂By∂z=−μ0ε0∂Ex∂t.\frac{\partial E_x}{\partial z} = -\frac{\partial B_y}{\partial t}, \qquad \frac{\partial B_y}{\partial z} = -\mu_0\varepsilon_0\frac{\partial E_x}{\partial t}. Combine them into a single equation for ExE_x, identify the speed of the shapes f(z−vt)f(z - vt) that solve it, and compute that speed from your ε0\varepsilon_0 and the given μ0\mu_0. ((c)) Treat sunlight at the Earth as a plane wave travelling along +z+z with E=E0cos⁡(kz−ωt)x^\mathbf{E} = E_0\cos(kz-\omega t)\hat{\mathbf{x}}. Find E0E_0 and the amplitude B0B_0, and state the direction of B\mathbf{B}. ((d)) A flat sail of area 100 m2100\,\mathrm{m^2}, taken as a perfect reflector, is held square-on to the sunlight at the Earth's distance. Find the force on it.

Browse the problem library