Question
The electric field associated with an electromagnetic wave in vacuum is given by $$E = \hat i40\cos \left( {kz - 6 \times {{10}^8}t} \right),$$ where $$E,z$$ and $$t$$ are in $$volt/m,$$ metre and second respectively. The value of wave vector $$k$$ is
A.
$$2\,{m^{ - 1}}$$
B.
$$0.5\,{m^{ - 1}}$$
C.
$$6\,{m^{ - 1}}$$
D.
$$3\,{m^{ - 1}}$$
Answer :
$$2\,{m^{ - 1}}$$
Solution :
Electromagnetic wave equation is given by
$$E = {E_0}\cos \left( {kz - \omega t} \right)\,......\left( {\text{i}} \right)$$
Speed of electromagnetic wave, $$v = \frac{\omega }{k}$$
Given, equation is
$$E = \hat i40\cos \left( {kz - 6 \times {{10}^8}t} \right)\,.....\left( {{\text{ii}}} \right)$$
Comparing Eqs. (i) and (ii), we get
$$\omega = 6 \times {10^8}$$
and $${E_0} = 40\,\hat i$$
Here, wave factor, $$k = \frac{\omega }{v} = \frac{{6 \times {{10}^8}}}{{3 \times {{10}^8}}} = 2\,{m^{ - 1}}$$