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The electric field of an electromagnetic wave in free space is given by $\vec E$$=10 cos (10^7t+kx)$$\hat j$ $volt/m $ where $t$ and $x$ are in seconds and metres respectively. It can be inferred that
$(1)$ the wavelength $\lambda$ is $188.4\, m.$
$(2)$ the wave number $k$ is $0.33\,\, rad/m.$
$(4)$ the wave is propagating along $+x$ direction.
Which one of the following pairs of statements is correct ?
$(3)$ and $(4)$
$(1)$ and $(2)$
$(2)$ and $(3)$
$(1)$ and $(3)$
Solution
As given
$E=10 \cos \left(10^{7} t+k x\right)………(i)$
Comparing it with standard equation of e.m. wave,
$E=E_{0} \cos (\omega t+k x)………(ii)$
Amplitude $E_{0}=10\, \mathrm{V} / \mathrm{m}$ and $\omega=10^{7} \,\mathrm{rad} / \mathrm{s}$
$\because \quad c=u \lambda=\frac{\omega \lambda}{2 \pi}$
or $\quad \lambda=\frac{2 \pi c}{\omega}=\frac{2 \pi \times 3 \times 10^{8}}{10^{7}}=188.4 \,\mathrm{m}$
Also, $c=\frac{\omega}{k}$ or $k=\frac{\omega}{c}=\frac{10^{7}}{3 \times 10^{8}}=0.033$
The wave is propagating along $y$ direction.
