Question

The rms value of electric field of light coming from sun is \(720 \mathrm{~N} / \mathrm{c}\). The average total energy density of electromagnetic wave is (A) \(6.37 \times 10^{-9} \mathrm{~J} / \mathrm{m}^{3}\) (B) \(4.58 \times 10^{-6} \mathrm{~J} / \mathrm{m}^{3}\) (C) \(81.35 \times 10^{-12} \mathrm{~J} / \mathrm{m}^{3}\) (A) \(3.3 \times 10^{-3} \mathrm{~J} / \mathrm{m}^{3}\)

Short answer

The average total energy density of the electromagnetic wave is (B) \(4.58 \times 10^{-6} \frac{J}{m^3}\).

Step-by-step solution

First, remember that the rms value of electric field (E) is given to us: \( E_{rms} = 720 \frac{N}{C} \). #Step 2: Calculate the magnetic field (B) rms value#

Now, we need to find the corresponding rms value of the magnetic field (B) in the electromagnetic wave. We can use the equation: \( B_{rms} = \frac{E_{rms}}{c} \), where c is the speed of light \((3 \times 10^8 \frac{m}{s})\). So, \[ B_{rms} = \frac{720 \frac{N}{C}}{3 \times 10^8 \frac{m}{s}} = 2.4 \times 10^{-6} \frac{T (Tesla)}{m} \] #Step 3: Calculate average energy density#

Next, let's calculate the average total energy density (u) of the electromagnetic wave. The formula for energy density is: \(u = \frac{1}{2} \epsilon_0 E_{rms}^2 + \frac{1}{2} \frac{B_{rms}^2}{\mu_0}\), where \(\epsilon_0\) is the vacuum permittivity \((8.854 \times 10^{-12} \frac{C^2}{N m^2})\) and \(\mu_0\) is the vacuum permeability \((4 \pi \times 10^{-7} \frac{T m}{A})\). #Step 4: Insert values and calculate the answer#

Let's plug in the given values and calculate u: \[u = \frac{1}{2} \times (8.854 \times 10^{-12} \frac{C^2}{N m^2}) \times (720 \frac{N}{C})^2 + \frac{1}{2} \times \frac{(2.4 \times 10^{-6} \frac{T}{m})^2}{(4 \pi \times 10^{-7} \frac{T m}{A})}\] After calculating, we get: \[u = 4.58 \times 10^{-6} \frac{J}{m^3}\] So the correct answer is (B) \(4.58 \times 10^{-6} \frac{J}{m^3}\).

Key concepts

RMS Electric Field

When we talk about the electric field in electromagnetic waves, especially in terms of the root mean square (rms) value, we're discussing a kind of average electric field strength. The arems value of an electric field (E_{rms}) is essentially the square root of the average of the squares of all instantaneous field values.
This calculation helps us deal with electric fields that vary over time, like those of light from the sun.

• The rms value smooths out the highs and lows of an oscillating wave.
• In this exercise, the electric field rms value is given as \(720 \, \mathrm{N/C}\).

This value provides a useful summary statistic for the varying electric field, and it's instrumental in calculating other properties of electromagnetic waves, like energy density.

Magnetic Field Calculation

The calculation of the rms value of the magnetic field is necessary for determining the complete energy density of an electromagnetic wave. According to electromagnetic theory, the rms value of the magnetic field (B_{rms}) can be derived from the rms value of the electric field using the relationship:\[ B_{rms} = \frac{E_{rms}}{c} \]where \( c \) is the speed of light, \( 3 \times 10^8 \, \mathrm{m/s} \).
For this particular problem:

• We are given \( E_{rms} = 720 \, \mathrm{N/C} \).
• Plugging into the formula gives: \( B_{rms} = \frac{720}{3 \times 10^8} = 2.4 \times 10^{-6} \mathrm{T} \).

This calculation provides the magnetic component of the electromagnetic wave, crucial for assessing the wave's energy content.

Vacuum Permittivity and Permeability

Vacuum permittivity and permeability are fundamental physical constants that quantify the electric and magnetic properties of a vacuum. These constants are crucial for understanding how electromagnetic waves propagate in a vacuum:

• Vacuum permittivity (\(\epsilon_0\)): This describes how electric fields interact with the vacuum, with a value of \(8.854 \times 10^{-12} \, \mathrm{C^2/N\cdot m^2}\).
• Vacuum permeability (\(\mu_0\)): This describes how magnetic fields interact with the vacuum, with a value of \(4\pi \times 10^{-7} \, \mathrm{T\cdot m/A}\).

In the context of this exercise, these constants enter the calculation of the average energy density \(u\) of an electromagnetic wave:\[ u = \frac{1}{2} \epsilon_0 E_{rms}^2 + \frac{1}{2} \frac{B_{rms}^2}{\mu_0} \] Understanding these constants is essential for accurately evaluating the behavior of electromagnetic fields in a vacuum, which forms the basis for electromagnetic theory.