Question

Calculate the average value of the Poynting vector, \(S_{\text {ave }}\) for an electromagnetic wave having an electric field of amplitude \(100 . \mathrm{V} / \mathrm{m}\) a) What is the average energy density of this wave? b) How large is the amplitude of the magnetic field?

Short answer

Answer: The average energy density of the wave is approximately \(8.85 \times 10^{-9} \,\mathrm{J/m^3}\) and the amplitude of the magnetic field is approximately \(3.33 \times 10^{-7} \,\mathrm{T}\).

Step-by-step solution

Calculate the average value of the Poynting vector

The Poynting vector, \(\boldsymbol{S}\), represents the power, or the rate of energy flow, per unit area in an electromagnetic wave. It can be found using the formula: $$\boldsymbol{S} = \frac{1}{\mu_0}\boldsymbol{E} \times \boldsymbol{B}$$ Where \(\boldsymbol{E}\) and \(\boldsymbol{B}\) are the electric and magnetic fields, respectively, and \(\mu_0\) is the permeability of free space (\(4 \pi \times 10^{-7} \,\mathrm{Tm/A}\)). However, we want to find the average value of the Poynting vector, which can be obtained as follows: $$S_{\text{ave}} = \frac{1}{2} \epsilon_0 c |E_0|^2$$ Where \(\epsilon_0\) is the permittivity of free space (\(8.85 \times 10^{-12} \,\mathrm{F/m}\)), \(c\) is the speed of light (\(3 \times 10^8 \,\mathrm{m/s}\)), and \(E_0\) is the amplitude of the electric field. Given the electric field amplitude of \(100 \,\mathrm{V/m}\), we can find \(S_{\text{ave}}\).

Calculate the average energy density of the wave

The total energy density, \(u\), of an electromagnetic wave is given by the sum of the electric and magnetic energy densities: $$u = u_\text{E} + u_\text{B}$$ Since the electric and magnetic fields are in phase, their average energy densities are equal and can be calculated as follows: $$u_\text{E} = \frac{1}{2}\epsilon_0 |E_0|^2$$ We can now find the average energy density of the wave.

Calculate the amplitude of the magnetic field

To find the amplitude of the magnetic field, \(B_0\), we can use the relationship between the electric and magnetic fields in an electromagnetic wave: $$|E_0| = c |B_0|$$ Using the given value of \(E_0\) and the speed of light \(c\), we can find the amplitude of the magnetic field. Calculations:

Calculate the average value of the Poynting vector

Using the given values, we can calculate \(S_{\text{ave}}\): $$S_{\text{ave}} = \frac{1}{2} \times 8.85 \times 10^{-12} \times 3 \times 10^8 \times (100)^2$$ $$S_{\text{ave}} \approx 1.327 \times 10^{-3} \,\mathrm{W/m^2}$$

Calculate the average energy density of the wave

To calculate the average energy density of the wave, we can calculate \(u_\text{E}\) and then find the total energy density: $$u_\text{E} = \frac{1}{2} \times 8.85 \times 10^{-12} \times (100)^2$$ $$u_\text{E} \approx 4.425 \times 10^{-9} \,\mathrm{J/m^3}$$ Since \(u_\text{E} = u_\text{B}\), the total energy density \(u\) is: $$u = 2u_\text{E} = 2 \times 4.425 \times 10^{-9} \,\mathrm{J/m^3}$$ $$u \approx 8.85 \times 10^{-9} \,\mathrm{J/m^3}$$

Calculate the amplitude of the magnetic field

Using the given values, we can find the amplitude of the magnetic field: $$|B_0| = \frac{|E_0|}{c} = \frac{100}{3 \times 10^8}$$ $$|B_0| \approx 3.33 \times 10^{-7} \,\mathrm{T}$$ To summarize: a) The average energy density of the wave is approximately \(8.85 \times 10^{-9} \,\mathrm{J/m^3}.\) b) The amplitude of the magnetic field is approximately \(3.33 \times 10^{-7} \,\mathrm{T}.\)

Key concepts

Electromagnetic Wave

An electromagnetic wave is a fascinating phenomenon where electric and magnetic fields oscillate in harmony, traveling through space at the speed of light. These waves do not require a medium and can propagate through the vacuum of space. This is why we can receive light from the sun and stars across vast distances.

Electromagnetic waves are transverse in nature, which means that the oscillations of the electric and magnetic fields occur perpendicular to the direction the wave travels. The most well-known example of an electromagnetic wave is light, but these waves range from radio waves, microwaves, and infrared, all the way to visible light, ultraviolet, X-rays, and gamma rays.

• The frequencies of electromagnetic waves vary, defining the type of wave and its properties.
• Their speed in a vacuum is constant, given by the speed of light, which is approximately \(3 \times 10^8 \text{ m/s}\).

The electric and magnetic fields are always in phase and oscillate perpendicular to each other, creating a dynamic system that efficiently transmits energy.

Energy Density

Energy density in the context of electromagnetic waves refers to the energy stored in the electric and magnetic components of the wave per unit volume. It provides insight into how much energy is transported through space by the wave.

The total energy density \(u\) is the sum of the electric energy density \(u_E\) and the magnetic energy density \(u_B\). In an electromagnetic wave, these energy densities are equal on average, leading to a simple relationship:

• The electric energy density \(u_E = \frac{1}{2} \epsilon_0 |E_0|^2\).
• The magnetic energy density \(u_B = \frac{1}{2 \mu_0} |B_0|^2\).
• Total energy density \(u = u_E + u_B\) is often simply \(2u_E\) because \(u_E = u_B\).

Understanding energy density is crucial because it tells us how much energy is available from the wave and can be useful in applications like solar power calculations or understanding radiation exposure.

Electric Field

The electric field in electromagnetic waves is a vector field that represents the force experienced by a charge within the field. Its role in electromagnetic waves is to oscillate and generate magnetic fields, contributing to the wave's propagation.

For electromagnetic waves, the amplitude of the electric field \(E_0\) means the maximum strength of the electric field wave as it oscillates. This amplitude is fundamental for determining the wave's characteristics, such as its power and energy density:

• The electric field in a wave oscillates perpendicular to the magnetic field and propagation direction.
• Its strength and direction change with time but maintain a consistent phase with the magnetic field.
• In our example, the amplitude of \(100 \text{ V/m}\) was used to derive the Poynting vector and energy density.

The concept of the electric field is intertwined with numerous fundamental physics principles, and its dynamic nature in waves is pivotal for a variety of technologies, including communication systems and electromagnetic applications.

Magnetic Field

In electromagnetic waves, the magnetic field works hand-in-hand with the electric field. It's a vector field that signifies the influence a moving charge will experience due to the wave's passage.

The relationship between the electric field \(E\) and the magnetic field \(B\) in an electromagnetic wave is expressed by \(|E_0| = c |B_0|\), where \(c\) is the speed of light. This gives us a straightforward method to calculate the magnetic field's amplitude given the electric field's amplitude:

• The magnetic field oscillates in phase with and perpendicular to the electric field.
• It plays a critical role in determining the wave's characteristics.
• In our example, from the electric field amplitude of \(100 \text{ V/m}\), the magnetic field amplitude was calculated to be approximately \(3.33 \times 10^{-7} \text{ T}\).

Understanding the magnetic field in electromagnetic waves helps us tap into how energy is transferred and used, further aiding in the development of technologies like MRI in healthcare and in deeper space explorations.