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 directly proportional to the square of the electric field \( E \). The energy density is defined as:

\[ U = \frac{\epsilon_0 E^2}{2} \]

Step 1: Given that both cylinders possess identical energy, we establish the equality:

\[ U_1 = U_2 \]

Step 2: Since energy is proportional to the square of the electric field, we infer:

\[ E_1^2 \propto E_2^2 \]

For the second cylinder, its diameter is halved, resulting in a fourfold reduction in its area.

Step 3: Consequently, to counterbalance the diminished area, the electric field must decrease by a factor of 2.

Step 4: Therefore, the new electric field will be \( \frac{1}{2} \) of the original. This yields the new electric field as:

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

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