Question

Due to the presence of an em-wave whose electric component is given by \( E = 100 \sin(\omega t - kx) \, \text{N/C} \), a cylinder of length 200 cm holds a certain amount of em-energy inside it. If another cylinder of the same length but half the diameter of the previous one holds the same amount of em-energy, the magnitude of the electric field of the corresponding em-wave should be modified as:

• A. \[ 200 \sin(\omega t - kx) \, \text{N/C}^{-1} \]
• B. \[ 25 \sin(\omega t - kx) \, \text{N/C}^{-1} \]
• C. \[ 50 \sin(\omega t - kx) \, \text{N/C}^{-1} \]
• D. \[ 400 \sin(\omega t - kx) \, \text{N/C}^{-1} \]

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:

\[ E_2 = 50 \sin(\omega t - kx) \, \text{N/C}^{-1} \]

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