Question

Which of the following have zero average value in a plane electromagnetic wave? (A) Electric energy (B) Magnetic energy (C) Electric field (D) None of these.

Short answer

The average value of the electric field in a plane electromagnetic wave is zero (C).

Step-by-step solution

1. Electric Energy

The electric energy density of an electromagnetic wave is 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 amplitude. The electric field varies sinusoidally in time, as \(E(t) = E_0 \sin(\omega t)\). To find the average value of electric energy in one time period, we can integrate over time: \(\bar{u}_E = \frac{1}{T}\int_0^{T} \frac{1}{2}\epsilon_0 (E_0 \sin(\omega t))^2 dt \)

2. Magnetic Energy

The magnetic energy density of an electromagnetic wave is given by: \(u_B = \frac{1}{2\mu_0} B^2\) Where \(\mu_0\) is the permeability of free space, and \(B\) is the magnetic field amplitude. The magnetic field also varies sinusoidally in time, as \(B(t) = B_0 \sin(\omega t)\). To find the average value of magnetic energy in one time period, we can integrate over time: \(\bar{u}_B = \frac{1}{T}\int_0^{T} \frac{1}{2\mu_0} (B_0 \sin(\omega t))^2 dt \)

3. Electric field

The electric field in a plane electromagnetic wave varies sinusoidally as \(E(t) = E_0 \sin(\omega t)\). To find the average value of the electric field in one time period, we can integrate over time: \(\bar{E} = \frac{1}{T}\int_0^{T} E_0 \sin(\omega t) dt \)

4. Find the average values of each quantity

For electric energy: \(\bar{u}_E = \frac{1}{T}\int_0^{T} \frac{1}{2}\epsilon_0 (E_0 \sin(\omega t))^2 dt = \frac{1}{2} \epsilon_0 E_0^2 \cdot\frac{1}{2}\) (as the average value of \(\sin^2(\omega t)\) over one period is \(\frac{1}{2}\)) For magnetic energy: \(\bar{u}_B = \frac{1}{T}\int_0^{T} \frac{1}{2\mu_0} (B_0 \sin(\omega t))^2 dt = \frac{1}{2\mu_0} B_0^2 \cdot\frac{1}{2}\) (as the average value of \(\sin^2(\omega t)\) over one period is \(\frac{1}{2}\)) For electric field: \(\bar{E} = \frac{1}{T}\int_0^{T} E_0 \sin(\omega t) dt = E_0 \cdot 0\) (as the average value of \(\sin(\omega t)\) over one period is 0) From the results above, we can see that the electric energy and magnetic energy have non-zero average values in a plane electromagnetic wave, while the average value of the electric field is zero. Therefore, the correct answer is: (C) Electric field

Key concepts

Electric Field

An electric field is a region where an electric charge experiences a force. It is represented by the symbol \(E\). In a plane electromagnetic wave, the electric field varies sinusoidally. This means it oscillates up and down over time.
The equation for an electric field in such a wave can be written as \(E(t) = E_0 \sin(\omega t)\), where:

• \(E_0\) is the peak value or the amplitude of the electric field.
• \(\omega\) is the angular frequency, determining how fast the wave oscillates.

The average value of an electric field over a full cycle is zero. Since sine waves oscillate equally above and below zero, they average out to zero over a complete period.

Magnetic Energy

Magnetic energy in an electromagnetic wave is derived from the magnetic field. This field also oscillates sinusoidally just like the electric field. The magnetic energy density, which measures the energy stored in the magnetic field per unit volume, is given by:\[u_B = \frac{1}{2\mu_0} B^2\]Here,

• \(\mu_0\) represents the permeability of free space, indicating how a magnetic field permeates through a vacuum.
• \(B\) is the magnetic field's amplitude.

The magnetic energy changes over time as \(B(t)\) varies, and the average value of this energy over a complete oscillation cycle is not zero. Therefore, magnetic energy contributes continuously to wave propagation.

Electric Energy

Electric energy in an electromagnetic wave stems from the electric field. Its energy density, a measure of stored energy per unit volume, can be calculated with:\[u_E = \frac{1}{2}\epsilon_0 E^2\]Where:

• \(\epsilon_0\) is the permittivity of free space, dictating how an electric field affects and is affected by a vacuum.
• \(E\) is the amplitude of the electric field.

As the field oscillates, the stored energy also varies, similar to the magnetic field but stored electrically. Over one full cycle, the average electric energy is not zero, so it equally contributes to energy dissemination in a wave.

Permeability of Free Space

This concept, often symbolized as \(\mu_0\), indicates how well a magnetic field can penetrate through a vacuum. It is a fundamental parameter in the equations governing electromagnetic theory, essential in calculating magnetic energy density.

Permeability of free space is crucial for understanding how electromagnetic waves travel through space. It allows us to understand the relationship between electric and magnetic fields. The value of \(\mu_0\) is a universal constant and plays a vital role in Maxwell's equations, which are the foundation of electromagnetism.

Permittivity of Free Space

Permittivity of free space, like permeability, is a foundational constant in electromagnetism. Represented by \(\epsilon_0\), it measures the ability of a vacuum to support an electric field and retains energy electrically in an electromagnetic wave.

\(\epsilon_0\) is integral in the formula for electric energy density and reflects how electric fields interact within space. This permittivity, a constant, is critical for calculating electric forces and fields in a vacuum.

• Permittivity and permeability constants help understand the speed of light through a vacuum, defined by \(c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}\).

These constants create a deeper insight into the interaction between electric and magnetic components of an electromagnetic wave.