Question

The energy associated with electric field is E and with magnetic field is B for an electromagnetic wave in free space is given by( \(∈_0\)-permittivity of free space \(μ_0\)-permeability of free space) 

• A. \(U_E=\frac{E^2}{2∈_0},U_B=\frac{μ_0B^2}{2}\)
• B. \(U_E=\frac{∈E^2}{2},U_B=\frac{μ_0B^2}{2}\)
• C. \(U_E=\frac{E^2}{2∈_0},U_B=\frac{B^2}{2μ_0}\)
• D. \(U_E=\frac{∈_0E^2}{2},U_B=\frac{B^2}{2μ_0}\)

Answer 1

The Correct Option is D

To determine the expressions for the energy associated with the electric field \(E\) and magnetic field \(B\) in an electromagnetic wave in free space, we utilize the following well-known formulas from electromagnetic theory.

The energy density (\(U_E\)) of the electric field in free space can be given by:

\(U_E = \frac{1}{2} \epsilon_0 E^2\)

where \( \epsilon_0 \) is the permittivity of free space, and \(E\) is the electric field magnitude.

Similarly, the energy density (\(U_B\)) of the magnetic field in free space is given by:

\(U_B = \frac{1}{2} \frac{B^2}{\mu_0}\)

where \( \mu_0 \) is the permeability of free space, and \(B\) is the magnetic field magnitude.

The correct answer matches these standard expressions for energy densities in electromagnetic theory. Hence, the correct option is:

\(U_E = \frac{\epsilon_0 E^2}{2}, \, U_B = \frac{B^2}{2\mu_0}\)

Let's rule out the other options:

• \(U_E = \frac{E^2}{2\epsilon_0}, U_B = \frac{\mu_0 B^2}{2}\): This option is incorrect as the formula for \(U_E\) does not correctly incorporate permittivity \( \epsilon_0\), which should be in the numerator.
• \(U_E = \frac{\epsilon E^2}{2}, U_B = \frac{\mu_0 B^2}{2}\): This option uses \( \epsilon \), not \( \epsilon_0\), which is crucial as \( \epsilon_0\) is the electric constant or permittivity of free space.
• \(U_E = \frac{E^2}{2\epsilon_0}, U_B = \frac{B^2}{2\mu_0}\): While the formula for \(U_B\) is correct, the formula for \(U_E\) does not properly incorporate the permittivity of free space, \( \epsilon_0\), which should be in the expression.

Conclusion: The correct expressions representing the energy densities of the electric and magnetic fields are crucial in understanding electromagnetic waves in free space.