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An electromagnetic wave travelling in the $x-$ direction has frequency of $2 \times 10^{14}\,Hz$ and electric field amplitude of $27\,Vm^{-1}$ . From the options given below, which one describes the magnetic field for this wave ?
$\vec B\,(x\,,\,t) = (3 \times {10^{ - 8}}\,T)\,\hat j \;\sin \,[2\pi \,(1.5 \times {10^{ - 8}}\,x\, - \,2 \times {10^{14}}\,t)]$
$\vec B\,(x\,,\,t) = (9 \times {10^{ - 8}}\,T)\,\hat i\; \sin \,[2\pi \,(1.5 \times {10^{ - 8}}\,x\, - \,2 \times {10^{14}}\,t)]$
$\vec B\,(x\,,\,t) = (9 \times {10^{ - 8}}\,T)\,\hat j\;\sin \,[(1.5 \times {10^{ - 6}}\,x\, - \,2 \times {10^{14}}\,t)]$
$\vec B\,(x\,,\,t) = (9 \times {10^{ - 8}}\,T)\,\hat k \;\sin \,[2\pi \,(1.5 \times {10^{ - 6}}\,x\, - \,2 \times {10^{14}}\,t)]$
Solution
As we know,
$B_{0}=\frac{E_{0}}{C}=\frac{27}{3 \times 10^{8}}=9 \times 10^{-8}$ $tesla$
Oscillation of $B$ can be only along $\hat{\jmath}$ or $\hat{k}$ direction.
$\omega=2 \pi f=2 \pi \times 2 \times 10^{14} \,\mathrm{Hz}$
$\therefore \bar{B}(x, t)=\left(9 \times 10^{-8} T\right) \hat{k} \sin [2 \pi(1)\left.1.5 \times 10^{-6} \times-2 \times 10^{4} t\right)$
