Question

Due to presence of an em-wave whose electric component is given by \( E = 100 \sin(\omega t - kx) \, NC^{-1} \), 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:

• A. \( 200 \sin(\omega t - kx) \, NC^{-1} \)
• B. \( 25 \sin(\omega t - kx) \, NC^{-1} \)
• C. \( 50 \sin(\omega t - kx) \, NC^{-1} \)
• D. \( 400 \sin(\omega t - kx) \, NC^{-1} \)

Hint:

In problems involving energy conservation, remember that energy is proportional to the square of the electric field. A change in the physical dimensions of the setup requires adjustments in the electric field.

Answer 1

The Correct Option is C

The energy of an electromagnetic wave is proportional to the square of the electric field \( E \), and the energy density is given by: \[ U = \frac{\epsilon_0 E^2}{2} \] 

Step 1: Since both cylinders contain the same amount of energy, we have: \[ U_1 = U_2 \] 

Step 2: The energy is proportional to the square of the electric field: \[ E_1^2 \propto E_2^2 \] For the second cylinder, the diameter is half, which reduces the area by a factor of 4. 

Step 3: Therefore, the electric field should decrease by a factor of 2 to compensate for the reduced area. 

Step 4: Thus, the new electric field will be \( \frac{1}{2} \) of the original, making the new electric field: \[ E_2 = 50 \sin(\omega t - kx) \, NC^{-1} \]

Final Conclusion: The modified electric field is \( 50 \sin(\omega t - kx) \, NC^{-1} \), which corresponds to Option (3).