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  2. Solve! - Electromagnetic Waves - JEE Main
# Engineering

The magnetic field of an electromagnetic wave is given by:

\vec{B}=1.6 \times 10^{-6} \cos \left ( 2 \times 10^{7}z + 6 \times 10^{15}t \right )\left ( 2\hat{i} +\hat{j} \right )\frac{Wb}{m^{2}}

The associated electric field will be :

 

  • Option 1)

    \vec{E}=4.8 \times 10^{2} cos \left ( 2 \times 10^{7}z-6 \times 10^{15}t \right )\left ( 2 \hat{i}+\hat{j} \right )\frac{V}{m}

  • Option 2)

    \vec{E}=4.8 \times 10^{2} cos \left ( 2 \times 10^{7}z-6 \times 10^{15}t \right )\left ( -2 \hat{j}+\hat{i} \right )\frac{V}{m}

  • Option 3)

    \vec{E}=4.8 \times 10^{2} cos \left ( 2 \times 10^{7}z-6 \times 10^{15}t \right )\left ( -\hat{i}+2\hat{j} \right )\frac{V}{m}

  • Option 4)

    \vec{E}=4.8 \times 10^{2} cos \left ( 2 \times 10^{7}z-6 \times 10^{15}t \right )\left ( \hat{i}+2\hat{j} \right )\frac{V}{m}

 

Answers (1)
S solutionqc

\vec{B}=1.6 \times10^{-6} \cos \left ( Kz +wt \right )\left ( 2\hat{i}+\hat{j} \right )

K=2 \times 10^{7}, w=6\times 10^{15}

B_{0}=1.6 \times 10^{-6}\times \sqrt{5}

so use E_{0}=CB_{0}

so E_{0}=1.6 \times 10^{-6}\times 3 \times 10^{8} \times \sqrt{5}

E_{0}=A.8 \times 10^{2} \sqrt{5}

& \vec{E} \times \vec{B} given direcction of wave

\vec{B} is along \left ( 2 \hat{i} + \hat{j} \right )

and wave is along -\hat{k}

so \vec{E}\times \left ( 2\hat{i} + \hat{j} \right )

should give -\hat{k}

option (1) \rightarrow \left ( 2\hat{i} + \hat{j} \right )\times \left ( 2\hat{i} + \hat{j} \right ) \Rightarrow 2\hat{k} -2 \hat{k}\Rightarrow 0

her \vec{E} & \vec{B} are parallel 

So option (1) is wrong 

option (2) \rightarrow E=E_{0} \cos \left ( Kz-wt \right )

But it should be E=E_{0} \cos \left ( Kz+wt \right )

so option 2 is wrong

option (3) \vec{E} \times \vec{B}

\left ( -\hat{i} +2 \hat{j} \right ) \times \left ( 2 \hat{i} +\hat{j} \right ) \Rightarrow -1\hat{k}-4\hat{k} \Rightarrow -5\hat{k}

direction is along -\hat{k}

So optiion (3) is correct

Option(4) \vec{E} \times \vec{B}\left ( \hat{i} -2 \hat{j} \right ) \times \left ( 2 \hat{i} +\hat{j} \right ) \Rightarrow \hat{k}+4\hat{k} \Rightarrow 5\hat{k}

but wave is along -k

so option (4) is wrong

 


Option 1)

\vec{E}=4.8 \times 10^{2} cos \left ( 2 \times 10^{7}z-6 \times 10^{15}t \right )\left ( 2 \hat{i}+\hat{j} \right )\frac{V}{m}

Option 2)

\vec{E}=4.8 \times 10^{2} cos \left ( 2 \times 10^{7}z-6 \times 10^{15}t \right )\left ( -2 \hat{j}+\hat{i} \right )\frac{V}{m}

Option 3)

\vec{E}=4.8 \times 10^{2} cos \left ( 2 \times 10^{7}z-6 \times 10^{15}t \right )\left ( -\hat{i}+2\hat{j} \right )\frac{V}{m}

Option 4)

\vec{E}=4.8 \times 10^{2} cos \left ( 2 \times 10^{7}z-6 \times 10^{15}t \right )\left ( \hat{i}+2\hat{j} \right )\frac{V}{m}

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