Problem 2196 - JEE Physics

Question

In a plane electromagnetic wave, the electric field oscillates sinusoidaly at a frequency of \(2.0 \times 10^{10} \mathrm{~Hz}\). if the peak value of electric field is \(60 \mathrm{Vm}^{-1}\) the average energy density (in \(\mathrm{Jm}^{-3}\) ) of the magnetic field of the wave will be (given \(\left.\mu_{0}=4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}\right)\) (A) \(2 \pi \times 10^{-7}\) (B) \((1 / 2 \pi) \times 10^{-7}\) (C) \(4 \pi \times 10^{-7}\) (D) \((1 / 4 \pi) \times 10^{-7}\)

Step-by-Step Solution

Answer

The short answer is: The average energy density of the magnetic field of the wave is (B) \(\frac{1}{2 \pi} \times 10^{-7} \mathrm{Jm}^{-3}\).

Step 1: Write down the given information

We are given: - Frequency of the wave (\(f\)): \(2.0 \times 10^{10} \mathrm{Hz}\) - Peak value of electric field (\(E_{\text{max}}\)): \(60 \mathrm{Vm}^{-1}\) - Vacuum permeability (\(\mu_{0}\)): \(4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}\)

Step 2: Using the relationship between electric and magnetic fields

In an electromagnetic wave, the electric field (\(E\)) and the magnetic field (\(B\)) are related by the speed of light in vacuum (\(c\)). The relationship is given by: \[E = cB\] From this equation, we can find the peak value of the magnetic field (\(B_{\text{max}}\)) as follows: \[B_{\text{max}} = \frac{E_{\text{max}}}{c}\]

Step 3: Calculate the speed of light

The speed of light in vacuum (\(c\)) is given by the following equation: \[c = \frac{1}{\sqrt{\mu_{0}\epsilon_{0}}}\] The vacuum permeability (\(\mu_{0}\)) is given, but we also need the vacuum permittivity (\(\epsilon_{0}\)). We can find this value using the equation: \[\epsilon_{0} = \frac{1}{\mu_{0}c^2}\] Using the value of \(\mu_{0}\), we get: \[\epsilon_{0} \approx 8.85\times10^{-12} \mathrm{C}^2 \mathrm{N}^{-1}\mathrm{m}^{-2}\] Now we can calculate the speed of light: \[c \approx 2.998 \times 10^8\, \mathrm{m\cdot s}^{-1}\]

Step 4: Calculate the peak value of the magnetic field

Now that we have the value of the speed of light, we can calculate the peak value of the magnetic field as follows: \[B_{\text{max}} = \frac{E_{\text{max}}}{c} = \frac{60}{2.998 \times 10^8} \approx 2 \times 10^{-7} \, \mathrm{T}\]

Step 5: Calculate the average magnetic energy density

The average magnetic energy density (\(u_B\)) is given by the following equation, \[u_B = \frac{1}{2}\mu_0 B_{\text{avg}}^2\] Since \(B_{\text{avg}}\) is half of the peak value, we have: \[B_{\text{avg}} = \frac{1}{2} B_{\text{max}} = \frac{1}{2}(2 \times 10^{-7}) = 1 \times 10^{-7} \, \mathrm{T}\] Now, we can calculate the average magnetic energy density: \[u_B = \frac{1}{2} (4 \pi \times 10^{-7})(1 \times 10^{-7})^2 =\frac{1}{2\pi} \times 10^{-7} \mathrm{Jm}^{-3}\] The answer is (B) \(\frac{1}{2 \pi} \times 10^{-7} \mathrm{Jm}^{-3}\).

Key Concepts

Magnetic Energy Density

The term "magnetic energy density" refers to how much magnetic energy is stored in a given volume of space. In formula terms, the magnetic energy density \( u_B \) can be calculated using the following equation:\[ u_B = \frac{1}{2} \mu_0 B_{\text{avg}}^2 \]where \( \mu_0 \) is the vacuum permeability and \( B_{\text{avg}} \) is the average value of the magnetic field. This concept helps us quantify the energy present in electromagnetic waves due to the magnetic field component.

When dealing with electromagnetic waves like light, both electric and magnetic fields carry energy. The total energy density is about the balanced interactions between these fields. Understanding how to determine the magnetic energy density is important for applications in physics and engineering, such as in the design of antennas and analyzing radio wave propagation.

Speed of Light

The speed of light \( c \) is a fundamental constant in physics, with a value of approximately \( 2.998 \times 10^8 \) meters per second. It represents the fastest speed at which information or matter can travel through space. In a vacuum, light travels at this maximum speed without any obstruction.

The mathematical relationship for the speed of light is given by:\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]In this equation, \( \mu_0 \) is the vacuum permeability and \( \epsilon_0 \) is the vacuum permittivity. These constants describe how electric and magnetic fields exist in a vacuum and are fundamental in determining the behavior of electromagnetic waves.
For electromagnetic waves, the speed of light provides the link between the electric field (\( E \)) and the magnetic field (\( B \)) through the equation:\[ E = cB \] This relationship illustrates the interdependence of electric and magnetic fields, showing how one can determine the magnitude of the other given the speed of light.

Vacuum Permeability

Vacuum permeability \( \mu_0 \), also known as the magnetic constant, is a measure of how a magnetic field propagates through a vacuum. It has a defined value of \( 4\pi \times 10^{-7} \) Tm/A.

The significance of \( \mu_0 \) lies in its role in the equation for the speed of light and its presence in formulas like the calculation of magnetic energy density. It represents the ability of a vacuum to conduct magnetic flux, impacting the interaction of magnetic fields in space.
In practical terms, vacuum permeability assists in understanding how electromagnetic waves behave not just in a vacuum but also in materials that can alter the permeability and influence the propagation of such waves. This is crucial in many scientific and engineering disciplines, especially in the design and analysis of electromagnetic systems.

Electric and Magnetic Fields Relationship

In electromagnetic waves, electric fields and magnetic fields are intrinsically linked. They oscillate perpendicular to each other and to the direction of the wave's propagation, forming an elegant wave structure.

The relationship between these fields is expressed in the equation:\[ E = cB \]where \( E \) is the electric field strength, \( B \) is the magnetic field strength, and \( c \) is the speed of light. This equation highlights the proportionality between the magnitudes of electric and magnetic fields in the wave.
Understanding this relationship is fundamental for grasping how electromagnetic waves propagate through space. This knowledge is applied widely in technologies such as radios, cell phones, and other wireless communication systems that exploit these field interactions to transmit signals over distances effectively. The synchronized dance of electric and magnetic fields is a key principle that enables all forms of modern electromagnetic communication.