Question
Which of the following is the correct equation for magnetic field as a function of $$x,$$ and $$t$$ where a plane electromagnetic wave propagating in the $$x$$-direction has a wavelength of $$5.0\,mm$$ ? The electric field is in the $$y$$-direction and its maximum value is $$30\,V{m^{ - 1}}.$$
A.
$$B = {10^{ - 3}}\sin \left[ {\frac{{2\pi }}{{5 \times {{10}^{ - 3}}}}\left( {ct - x} \right)} \right]$$
B.
$$B = {10^{ - 7}}\sin \left[ {\frac{{2\pi }}{{5 \times {{10}^{ - 3}}}}\left( {ct - x} \right)} \right]$$
C.
$$B = {10^{ - 5}}\sin \left[ {\frac{\pi }{{1.5 \times {{10}^{ - 3}}}}\left( {ct - x} \right)} \right]$$
D.
None of these
Answer :
$$B = {10^{ - 7}}\sin \left[ {\frac{{2\pi }}{{5 \times {{10}^{ - 3}}}}\left( {ct - x} \right)} \right]$$
Solution :
$$\eqalign{
& E = {E_0}\sin \omega \left( {t - \frac{x}{c}} \right);B = {B_0}\sin \omega \left( {t - \frac{x}{c}} \right) \cr
& {I_d} = {I_0}\sin \left( {\omega t + \frac{\pi }{2}} \right) = I\,{\text{and}}\,\omega = 2\pi v = \frac{{2\pi }}{\lambda }c \cr
& {E_0} = 30\sin \left[ {\frac{{2\pi }}{{5 \times {{10}^{ - 3}}}}\left( {ct - x} \right)} \right]\,{\text{and}}\,{B_0} = \frac{{{E_0}}}{C} = {10^{ - 7}}T, \cr
& B = {B_0}\sin \left[ {\frac{{2\pi }}{\lambda }\left( {ct - x} \right)} \right] \cr
& = {10^{ - 7}}\sin \left[ {\frac{{2\pi }}{{5 \times {{10}^{ - 3}}}}\left( {ct - x} \right)} \right] \cr} $$