Due to presence of an em-wave whose electric component is given by $E=100 \sin (\omega t-k x) \mathrm{NC}^{-1}$ a cylinder of length 200 cm holds certain amount of em-energy inside it. If another cylinder of same length but half diameter than previous one holds same amount of em-energy, the magnitude of the electric field of the corresponding em-wave should be modified as

<p>We know,</p> <p>Energy density $= {1 \over 2}{\varepsilon _0}{E^2}C$</p> <p>$$ \Rightarrow {\text{Energy} \over \text{Volume}} = {1 \over 2}{\varepsilon _0}{E^2}C$$</p> <p>$\Rightarrow \text{Energy}= {1 \over 2}{\varepsilon _0}E_c^2x\,vol$</p> <p>Given that, ${\left( {\text{Energy}} \right)_1} = {\left( {\text{Energy}} \right)_2}$</p> <p>$$ \Rightarrow {1 \over 2}{\varepsilon _0}E_1^2cx\pi R_1^2{L_1} = {1 \over 2}{\varepsilon _0}E_2^2cx\pi R_2^2{L_2}$$</p> <p>$\Rightarrow E_1^2R_1^2{L_1} = E_2^2R_2^2{L_1}\,(as\,{L_1} = {L_2})$</p> <p>$\Rightarrow E_1^2R_1^2 = E_2^2R_2^2$</p> <p>$\Rightarrow {E_1}{R_1} = {E_2}{R_2}$</p> <p>$$ \Rightarrow {E_1}{R_1} = {E_2}\left( {{{{R_1}} \over 2}} \right)\,\left( {As\,{R_2} = {{{R_1}} \over 2}} \right)$$</p> <p>$\Rightarrow {E_2} = 2{E_1} = 2 \times 100$</p> <p>$\Rightarrow {E_2} = 200\,N/C$</p> <p>Hence, option (3) is correct.</p>