Question:
In a plane electromagnetic wave, \(U_E\) and \(U_B\) are average energy densities of electric field and magnetic field respectively, then
In a plane electromagnetic wave, \(U_E\) and \(U_B\) are average energy densities of electric field and magnetic field respectively, then
Show Hint
For an electromagnetic wave,
\[
E=cB.
\]
As a result,
\[
U_E=\frac{1}{2}\varepsilon_0E^2
=
\frac{B^2}{2\mu_0}
=
U_B.
\]
Thus, the electric and magnetic fields contribute equally to the total energy density.
Updated On: Jun 26, 2026
- \(U_E=\dfrac{U_B}{2}\)
- \(U_E=2U_B\)
- \(U_E=U_B\)
- \(U_E\neq U_B\)
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The Correct Option is C
Solution and Explanation
Step 1: Write the expression for electric energy density.
The energy density associated with the electric field is \[ u_E=\frac{1}{2}\varepsilon_0E^2. \] The average electric energy density is therefore \[ U_E=\frac{1}{2}\varepsilon_0E_{\text{rms}}^2. \]
Step 2: Write the expression for magnetic energy density.
The energy density associated with the magnetic field is \[ u_B=\frac{B^2}{2\mu_0}. \] Hence, the average magnetic energy density is \[ U_B=\frac{B_{\text{rms}}^2}{2\mu_0}. \]
Step 3: Use the electromagnetic wave relation.
For an electromagnetic wave, \[ E=cB, \] where \[ c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}. \] Substituting \[ E^2=c^2B^2 = \frac{B^2}{\mu_0\varepsilon_0} \] into the expression for \(U_E\), \[ U_E = \frac{1}{2}\varepsilon_0 \left( \frac{B^2}{\mu_0\varepsilon_0} \right). \] \[ U_E = \frac{B^2}{2\mu_0}. \] But \[ U_B=\frac{B^2}{2\mu_0}. \] Therefore, \[ U_E=U_B. \]
Step 4: Physical interpretation.
In an electromagnetic wave, energy is equally shared between the electric field and magnetic field.
Hence, the electric energy density and magnetic energy density are always equal.
Step 5: Final conclusion.
Therefore, \[ \boxed{U_E=U_B} \] Hence, the correct option is \[ \boxed{(3)} \]
The energy density associated with the electric field is \[ u_E=\frac{1}{2}\varepsilon_0E^2. \] The average electric energy density is therefore \[ U_E=\frac{1}{2}\varepsilon_0E_{\text{rms}}^2. \]
Step 2: Write the expression for magnetic energy density.
The energy density associated with the magnetic field is \[ u_B=\frac{B^2}{2\mu_0}. \] Hence, the average magnetic energy density is \[ U_B=\frac{B_{\text{rms}}^2}{2\mu_0}. \]
Step 3: Use the electromagnetic wave relation.
For an electromagnetic wave, \[ E=cB, \] where \[ c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}. \] Substituting \[ E^2=c^2B^2 = \frac{B^2}{\mu_0\varepsilon_0} \] into the expression for \(U_E\), \[ U_E = \frac{1}{2}\varepsilon_0 \left( \frac{B^2}{\mu_0\varepsilon_0} \right). \] \[ U_E = \frac{B^2}{2\mu_0}. \] But \[ U_B=\frac{B^2}{2\mu_0}. \] Therefore, \[ U_E=U_B. \]
Step 4: Physical interpretation.
In an electromagnetic wave, energy is equally shared between the electric field and magnetic field.
Hence, the electric energy density and magnetic energy density are always equal.
Step 5: Final conclusion.
Therefore, \[ \boxed{U_E=U_B} \] Hence, the correct option is \[ \boxed{(3)} \]
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