Question 147: EM waves

Question

A monochromatic beam of light has a frequency $$v = {3 \over {2\pi }} 	imes {10^{12}}Hz$$ and is propagating along the direction $${{\widehat i + \widehat j} \over {\sqrt 2 }}.$$ 
It is polarized along the $$\widehat k$$ direction. The acceptable form for the magnetic field is :

Accepted Answer

Given : Frequency $$f = {3 \over {2\pi }} 	imes {10^{12}}$$ Hz; direction $$ = {{\widehat i + \widehat j} \over {\sqrt 2 }}$$.
The beam is polarised along $${\widehat k}$$ direction.
Direction of magnetic field $$\overrightarrow B  = \left( {{{\widehat i + \widehat j} \over {\sqrt 2 }} 	imes \widehat k = {{ - \widehat j} \over {\sqrt 2 }} + {{\widehat i} \over {\sqrt 2 }}} ight)$$
Therefore, direction of $$\overrightarrow B  = \left( {{{\widehat i - \widehat j} \over {\sqrt 2 }}} ight)$$
Now from wave equation, we have
$$B = {B_0}\cos (kr - \omega t)$$
$$\omega  = 2\pi f = 2\pi  	imes {3 \over {2\pi }} 	imes {10^{12}}Hz = 3 	imes {10^{12}}Hz$$
Now, $$k = {{\widehat i + \widehat j} \over {\sqrt 2 }}$$;
$${B_0} = {{{E_0}} \over c}\left( {{{\widehat i - \widehat j} \over {\sqrt 2 }}} ight)$$
Therefore,
$$B = {{{E_0}} \over c}\left( {{{\widehat i - \widehat j} \over {\sqrt 2 }}} ight)\cos \left[ {{{10}^4}\left( {{{\widehat i + \widehat j} \over {\sqrt 2 }}} ight)\,.\,r - (3 	imes {{10}^{12}})t} ight]$$

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EM waves

A monochromatic beam of light has a frequency $v = {3 \over {2\pi }} \times {10^{12}}Hz and is propagating along the direction {{\widehat i + \widehat j} \over {\sqrt 2 }}. It is polarized along the \widehat k$ direction. The acceptable form for the magnetic field is :

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