Question

Light waves are propagating in vacuum. Derive the wave equation for the
associated magnetic field vector. On the basis of this equation, calculate the speed
of light.

Answer 1

Let the electric field and the magnetic field vector is along the y axis and along the z axis.

The linearly polarized plane wave is traveling along the x axis and let the speed of light is c.

$E =Eo​(x,t)j^​$ and $B =Bo​(x,t)k$

where x is the displacement along the x axis, t is the time.

As per the maxwell's equation for the space

$∇.E=0$ and $∇.B=0$

$∇×E=−\dfrac{∂t}{∂B}$ and $∇×B=\mu_{o}​\varepsilon_{o}​\dfrac{∂t}{∂E}$

now,

Now, equating the magnitudes of the faradays law

$\dfrac{∂x}{∂E}​=−\dfrac{∂t}{∂B}$

now taking the partial derivative

$\dfrac{∂x^2}{∂^2E}​=−\dfrac{∂t^2}{∂^2B}$

Similarly

Now from the the above

we know that the general equation of the wave travailing along the x axis

$\dfrac{∂ψ^2}{∂x}​=\dfrac{∂ψ^2}{ν^2∂x}$

From the second derivative of electric and magnetic field

$\mu_{o}​\varepsilon_{o}​=\dfrac{1}{c^2}$

$c=\dfrac{1}{\mu_{o}​\varepsilon_{o}}​​=\dfrac{1}{8.85×10^{−12}×4π×10^{−71}​}m/sec$

$c=2.97×10^8m/sec$