4. Helmholtz Equation Problem 4A: Demonstrate that both the electric and magnetic fields for a linearly polarized plane wave propagating in the 𝑧-direction in free space satisfy the Helmholtz equation.

Answer: Maxwell’s Equations: As seen in Lecture 8, Maxwell’s equations in free space (where there are no charges or currents, i.e., ρ=0 and J ̅=0) are: ∇∙E ̅=∇∙D ⃗=0 ∇∙H ̅=∇∙B ⃗=0 ∇×E ̅=-μ (dH ̅)/dt=∇×E ⃗=-(dB ⃗)/dt ∇×H ̅=ϵ (dE ̅)/dt=∇×H ⃗=(dD ⃗)/dt

Helmholtz Equation: As seen in Lecture 8, Helmholtz Equation for an electric field is given by ∇^2 E ⃗+β^2 E ⃗=0 where the phase constant, β=ω√μϵ=2π/λ. Given, as per this online source, the applies to any vector function, the following is also true (i.e. the Helmholtz Equation for a magnetic field is as follows): ∇^2 B ⃗+β^2 B ⃗=0

Linearly Polarised Plane Wave Propagating in the z-Direction: For a plane wave propagating in the z-direction, we can assume that the magnetic field and the electric field are in the x and y-direction (either way around). This is because we know, from Lecture 8, that the cross product between the magnetic field and the electric field gives the direction of propagation. We can visualise this using the right-hand rule, where each finger is perpendicular to each other: The right hand's index finger points in the direction of the electric field (E ⃗). We will assume the electric field is along the x-axis. The right hand's middle finger points in the direction of the magnetic field (B ⃗). We will assume the magnetic field is along the y-axis. The right hand’s thumb points in the direction of the wave's propagation, which we are told is the z-axis.

From this, as well as the notes from Lecture 9, we now know we can right the electric and magnetic fields as follows: E ̅(z,t)=|〖E_0〗_x a ̂_x | cos⁡(ωt+ϕ_0 ) B ̅(z,t)=|〖B_0〗_y a ̂_y | cos⁡(ωt+ϕ_0 ) where E_0 and B_0 are the amplitudes of the electric and magnetic fields, and ϕ_0 is assumed to be 0 for simplicity (Lecture 9).

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