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.

$\overrightarrow{E}=E_o (x,t) \hat{j}$ and $\overrightarrow{B}=B_o(x,t)\hat{k}$

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

As per the maxwell's equation for the space

$\nabla.E=0$ and $\nabla. B=0$

$\nabla\times E=-\dfrac{\partial B }{\partial t}$ and $\nabla\times B=\mu_o \epsilon_o\dfrac{\partial E}{\partial t}$

now,

Now, equating the magnitudes of the faradays law

$\dfrac{\partial E}{\partial x}=-\dfrac{\partial B}{\partial t}$

now taking the partial derivative

$\dfrac{\partial^2 E}{\partial x^2}=-\dfrac{\partial^2 B}{\partial t^2}$

Similarly

Now from the the above

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

$\dfrac{\partial\psi^2}{\partial x}=\dfrac{\partial \psi^2}{\nu^2\partial x}$

From the second derivative of electric and magnetic field

$\epsilon_o\times\mu_o=\dfrac{1}{c^2}$

$c=\dfrac{1}{\sqrt{\epsilon_o \mu_o}}=\dfrac{1}{8.85\times 10^{-12}\times 4\pi\times 10^{-7}} m/sec$

$c=2.97\times 10^{8}m/sec$