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For a plane electromagnetic wave propagating in x-direction ....

For a plane electromagnetic wave propagating in x-direction, which one of the following combination gives the correct possible directions for electric field (E) and magnetic field (B) respectively?

(1) $\hat j + \hat k, \hat j + \hat k$
(2) $-\hat j + \hat k, -\hat j - \hat k$
(3) $\hat j + \hat k, -\hat j - \hat k$
(4) $-\hat j + \hat k,-\hat j+\hat k$

Solution

The direction of propagation $\hat i$ is the same as the direction of $\vec E \times \vec B$.

Or, $\hat i = \frac {\vec E \times \vec B}{| \vec E \times \vec B |}$

Options (1) & (4)  have identical vectors making their cross product 0.

Option (3) has vectors in opposite directions making their cross product 0 again.

For Option (2), $( - \hat j + \hat k) \times ( - \hat j - \hat k) = \hat j \times \hat k - \hat k \times \hat j = 2\hat i$

$\therefore \frac{{\vec E \times \vec B}}{{|\vec E \times \vec B|}} = \frac{{2\hat i}}{2} = \hat i$

Sum of the coefficients in the expansion of $(x+y)^n$ ....
If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$