E → ( x , t ) = E ( x , t ) j → \overrightarrow{E}(x,t)=E(x,t)\overrightarrow{j} E ( x , t ) = E ( x , t ) j
B → ( x , t ) = B ( x , t ) k → \overrightarrow{B}(x,t)=B(x,t)\overrightarrow{k} B ( x , t ) = B ( x , t ) k
∇ ⋅ E → = 0 \nabla \cdot \overrightarrow{E}=0 ∇ ⋅ E = 0
∇ ⋅ B → = 0 \nabla \cdot \overrightarrow{B}=0 ∇ ⋅ B = 0
∇ × E → = − d B → d t \nabla \times \overrightarrow{E}=-\frac{d\overrightarrow{B}}{dt} ∇ × E = − d t d B
∇ × B → = μ 0 ϵ 0 d E → d t \nabla \times \overrightarrow{B}=\mu_0 \epsilon_0\frac{d\overrightarrow{E}}{dt} ∇ × B = μ 0 ϵ 0 d t d E
∇ × E ( x , t ) j → = d E d x k → \nabla \times E(x,t)\overrightarrow{j}=\frac{dE}{dx}\overrightarrow{k} ∇ × E ( x , t ) j = d x d E k
d E d x = − d B d t \frac{dE}{dx}=-\frac{dB}{dt} d x d E = − d t d B
d 2 B d x 2 = − d d x ( μ 0 ϵ 0 d E d t ) = − μ 0 ϵ 0 d d t d E d x = μ 0 ϵ 0 d d t d B d t = μ 0 ϵ 0 d 2 B d t 2 \frac{d^2B}{dx^2}=-\frac{d}{dx}(\mu_0\epsilon_0\frac{dE}{dt})=-\mu_0\epsilon_0\frac{d}{dt}\frac{dE}{dx}=\mu_0\epsilon_0\frac{d}{dt}\frac{dB}{dt}=\mu_0\epsilon_0\frac{d^2B}{dt^2} d x 2 d 2 B = − d x d ( μ 0 ϵ 0 d t d E ) = − μ 0 ϵ 0 d t d d x d E = μ 0 ϵ 0 d t d d t d B = μ 0 ϵ 0 d t 2 d 2 B
So, we have
d 2 B d x 2 = μ 0 ϵ 0 d 2 B d t 2 \frac{d^2B}{dx^2}=\mu_0\epsilon_0\frac{d^2B}{dt^2} d x 2 d 2 B = μ 0 ϵ 0 d t 2 d 2 B
The general wave equation for a wave ψ ( x , t ) \psi(x,t) ψ ( x , t ) traveling in the x x x direction with speed v v v
d 2 ψ d x 2 = 1 v 2 d 2 ψ d t 2 \frac{d^2\psi}{dx^2}=\frac{1}{v^2}\frac{d^2\psi}{dt^2} d x 2 d 2 ψ = v 2 1 d t 2 d 2 ψ
So,
μ 0 ϵ 0 = 1 v 2 → c = v = 1 μ 0 ϵ 0 = 2.997 ⋅ 1 0 8 m / s \mu_0\epsilon_0=\frac{1}{v^2}\to c=v=\frac{1}{\sqrt{\mu_0\epsilon_0}}=2.997\cdot10^8 m/s μ 0 ϵ 0 = v 2 1 → c = v = μ 0 ϵ 0 1 = 2.997 ⋅ 1 0 8 m / s
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