Problem 40 - University Physics with Modern Physics

Question

The most intense beam of light that can propagate through dry air must have an electric field whose maximum amplitude is no greater than the breakdown value for air: \(E_{\max }^{\operatorname{air}}=3.0 \cdot 10^{6} \mathrm{~V} / \mathrm{m},\) assuming that this value is unaffected by the frequency of the wave. a) Calculate the maximum amplitude the magnetic field of this wave can have. b) Calculate the intensity of this wave. c) What happens to a wave more intense than this?

Step-by-Step Solution

Answer

Answer: The maximum amplitude of the magnetic field of this wave is \(10^{-2} \frac{T}{m}\), the intensity of the wave is \(1.197 \times 10^{11} W/m^2\), and a more intense wave will cause ionization of air, leading to energy loss and localized heating.

Step 1: Calculate the maximum amplitude of the magnetic field

We know the maximum amplitude of the electric field in air \(E_{\max}^{\operatorname{air}}= 3.0 \cdot 10^6~\frac{V}{m}\) and the speed of light in vacuum \(c= 3.0 \cdot 10^8~\frac{m}{s}\). We can calculate the maximum amplitude of the magnetic field using the formula \(B_{\max} = \frac{E_{\max}}{c}\): $$B_{\max} = \frac{3.0 \cdot 10^6}{3.0\cdot 10^{8}} = 10^{-2} \frac{T}{m}$$

Step 2: Calculate the intensity of the wave

Now that we have the maximum amplitude of the electric field, we can calculate the intensity of the wave using the formula \(I = \frac{1}{2}c\epsilon_0E_{\max}^2\). The vacuum permittivity constant \(\epsilon_0 = 8.854 \times 10^{-12}~\frac{C^2}{N\cdot m^2}\). Plugging in the values, we get: $$I = \frac{1}{2} \cdot 3.0\cdot 10^8 \cdot 8.854 \times 10^{-12} \cdot (3.0 \cdot 10^6)^2$$ $$I = 1.197 \times 10^{11} W/m^2$$

Step 3: Explain what happens to a more intense wave

A wave with intensity greater than the calculated value will lead to a phenomenon called "optical breakdown" or "air breakdown" in the dry air. This is because if the electric field amplitude is greater than the breakdown value of air, it will cause the air to become ionized. Ionized air becomes a highly conductive plasma, which will absorb the energy of the light wave, causing localized heating and a loss of the light wave's intensity. To sum up: a) The maximum amplitude of the magnetic field of this wave can have is \(10^{-2} \frac{T}{m}\). b) The intensity of this wave is \(1.197 \times 10^{11} W/m^2\). c) A wave more intense than this will ionize the air, causing energy loss and localized heating.

Key Concepts

Electric Field Amplitude

Electric field amplitude represents the maximum strength of the electric field in an electromagnetic wave. For waves in dry air, there's a limit to this amplitude due to the breakdown voltage of air. The breakdown value for air is known to be \(E_{\max }^{\operatorname{air}}=3.0 \times 10^{6} \, \mathrm{V/m}\). This means that any wave with an electric field exceeding this amplitude will potentially cause ionization, known as air breakdown.
Understanding electric field amplitude is critical because it not only determines the wave's capability to propagate through air but also its interaction with substances. This value of the electric field dictates how strongly charged particles, such as electrons, are forced to move when exposed to the field.

• Electric field amplitude is measured in volts per meter (V/m).
• It signifies the wave's ability to exert force on charges.
• The maximum amplitude in dry air is restricted to prevent breakdown.

Magnetic Field Amplitude

Magnetic field amplitude is associated with the strength of the magnetic component of an electromagnetic wave. For dry air, it's closely related to the electric field amplitude through the speed of light, \(c\). By using the relationship \(B_{\max} = \frac{E_{\max}}{c}\), we can determine the magnetic field's amplitude when we know its electric counterpart.
In the given problem, since \(E_{\max}^{\operatorname{air}} = 3.0 \times 10^{6} \, \mathrm{V/m}\) and \(c = 3.0 \times 10^{8} \, \mathrm{m/s}\), we find \(B_{\max} = 10^{-2} \, \mathrm{T/m}\). This shows how a proportionate relationship exists between electric and magnetic fields in a propagating wave.

• The magnetic field amplitude is calculated using the electric field and the speed of light.
• It's expressed as the magnetic field strength per meter (T/m).
• Magnetic influences contribute to the overall energy and intensity of a wave.

Air Breakdown Phenomenon

Air breakdown occurs when the electric field of an electromagnetic wave exceeds the permissible threshold for air. When the electric field reaches or exceeds the breakdown voltage, the air becomes ionized. This means that electrons are stripped away from atoms, creating a plasma.
Once ionized, the air becomes a conductive medium, absorbing the energy of the wave. This results in a loss of wave intensity and can lead to localized heating effects. In practical terms, this is the threshold at which a light beam becomes unable to propagate further, as its energy is absorbed by the ionized air.

• Air breakdown involves ionization of air particles due to high electric fields.
• Post-ionization, the air conducts electricity, altering wave propagation.
• This phenomenon limits the maximum intensity of light through air.