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Chapter 22: Problem 41

Prove that, in an EM wave traveling in vacuum, the electric and magnetic energy densities are equal; that is, prove that,$$\frac{1}{2} \epsilon_{0} E^{2}=\frac{1}{2 \mu_{0}} B^{2}$$ at any point and at any instant of time.,

Short Answer

Expert verified
Answer: Yes, in an electromagnetic wave traveling in a vacuum, the electric and magnetic energy densities are equal at any given point and time.

Step by step solution

01

Identify the formulas for electric and magnetic energy densities

In an EM wave, the electric energy density (u_E) and magnetic energy density (u_B) are given by the following formulas: $$ u_E = \frac{1}{2}\epsilon_0 E^2 $$ and $$ u_B = \frac{1}{2\mu_0}B^2 $$ Where: - \(E\) is the electric field - \(B\) is the magnetic field - \(\epsilon_0\) is the vacuum permittivity (\(8.854\times10^{-12} F/m\)) - \(\mu_0\) is the vacuum permeability (\(4\pi\times10^{-7} Tm/A\))
02

Establish the relationship between \(E\) and \(B\) in an EM wave

In an EM wave propagating through a vacuum, the electric field (E) and magnetic field (\(B\)) are perpendicular to each other, and their magnitudes are related by the following equation: $$ E = cB $$ Where \(c\) is the speed of light in vacuum (\(3\times10^8 m/s\)).
03

Substitute the relationship between \(E\) and \(B\) into the formulas for energy densities

From the relationship above, we can express \(B\) in terms of \(E\). So we get: $$ B = \frac{E}{c} $$ Now, substitute this value of \(B\) into the magnetic energy density equation: $$ u_B = \frac{1}{2\mu_0}\left(\frac{E}{c}\right)^2 $$
04

Simplify the equation for magnetic energy density

Now, simplify the equation for magnetic energy density: $$ u_B = \frac{1}{2\mu_0}\frac{E^2}{c^2} $$
05

Prove that the energy densities are equal

Now, we just need to prove that \(u_E\) is equal to \(u_B\). From step 4, we have: $$ u_B = \frac{1}{2\mu_0}\frac{E^2}{c^2} $$ Note that \(c^2 = \frac{1}{\epsilon_0\mu_0}\), so we can write the equation for \(u_B\) as follows: $$ u_B = \frac{1}{2}\frac{E^2}{\epsilon_0} $$ Comparing the equations for \(u_E\) and \(u_B\), we can see that they are equal: $$ \frac{1}{2}\epsilon_0 E^2 = \frac{1}{2}\epsilon_0 E^2 $$ Thus, we can conclude that in an EM wave traveling in a vacuum, the electric and magnetic energy densities are equal at any given point and time.

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