Answer to Question #102848 in Optics for sakshi

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

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

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

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