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  4. The electric field in an electromagnetic wave is given by E =(50 NC -1) sin ω( t - x / c ) The energy contained in a cylinder of volume V is 5.5 × 10-12 J. The value of V is cm 3 (. given .ϵ0=8.8 × 10-12 C 2 N -1 m -2)

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Q. The electric field in an electromagnetic wave is given by $E =\left(50\, NC ^{-1}\right) \sin \omega( t - x / c )$
The energy contained in a cylinder of volume $V$ is $5.5 \times 10^{-12} \,J$. The value of $V$ is ____ $cm ^{3}$
$\left(\right.$ given $\left.\epsilon_{0}=8.8 \times 10^{-12} C ^{2} \,N ^{-1} \,m ^{-2}\right)$

JEE MainJEE Main 2021Electromagnetic Waves

Solution:

$E =50 \sin \left(\omega t -\frac{\omega}{ c } \cdot x \right)$
Energy density $=\frac{1}{2} \in_{0} E _{0}^{2}$
Energy for volume $V =\frac{1}{2} \in_{0} E _{0}^{2} \cdot V =5.5 \times 10^{-12}$
$\frac{1}{2} 8.8 \times 10^{-12} \times 2500 V =5.5 \times 10^{-12}$
$V =\frac{5.5 \times 2}{2500 \times 8.8}=.0005\, m ^{3}$
$=.0005 \times 10^{6}( c . m )^{3}$
$=500( c . m )^{3}$