Modern Physics 7 Question 71
1. A plane electromagnetic wave having a frequency $v=23.9$ GHz propagates along the positive $z$-direction in free space. The peak value of the electric field is $60 V / m$. Which among the following is the acceptable magnetic field component in the electromagnetic wave?
(a) $\mathbf{B}=2 \times 10^{7} \sin \left(0.5 \times 10^{3} z+1.5 \times 10^{11} t\right) \hat{\mathbf{i}}$
(Main 2019, 12 April II)
(b) $\mathbf{B}=2 \times 10^{-7} \sin \left(0.5 \times 10^{3} z-1.5 \times 10^{11} t\right) \hat{\mathbf{i}}$
(c) $\mathbf{B}=60 \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right) \hat{\mathbf{k}}$
(d) $\mathbf{B}=2 \times 10^{-7} \sin \left(1.5 \times 10^{2} x+0.5 \times 10^{11} t\right) \hat{\mathbf{j}}$
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Solution:
- In an electromagnetic wave, magnetic field and electric field are perpendicular to each other and both are also perpendicular to the direction of propagation of wave.
Now, given direction of propagation is along $z$-direction. So, magnetic field is in either $x$ or $y$ direction. Also, angular wave number for wave is
$$ k=\frac{2 \pi}{\lambda}=\frac{2 \pi \nu}{c}=\frac{2 \pi \times 23.9 \times 10^{9}}{3 \times 10^{8}} \approx 0.5 \times 10^{3} m^{-1} $$
and angular frequency $\omega$ for wave is
$$ \omega=2 \pi \nu=2 \pi \times 23.9 \times 10^{9} Hz=1.5 \times 10^{11} Hz $$
Magnitude of magnetic field is
$$ B _0=\frac{E _0}{c}=\frac{60}{3 \times 10^{8}}=2 \times 10^{-7} T $$
As the general equation of magnetic field of an electromagnetic wave propagating in $+z$-direction is given as,
$$ \mathbf{B}=B _0 \sin (k z-\omega t) \hat{\mathbf{i}} \text { or } \hat{\mathbf{j}} $$
Thus, substituting the values of $B _0, k$ and $\omega$, we get $\Rightarrow \mathbf{B}=2 \times 10^{-7} \sin \left(0.5 \times 10^{3} z-1.5 \times 10^{11} t\right) \hat{\mathbf{i}}$ or $\hat{\mathbf{j}}$
