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Chapter 31: Problem 40

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?

Short Answer

Expert verified
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 by step solution

01

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 \(B_{\max} = \frac{E_{\max}}{c}\): $$B_{\max} = \frac{3.0 \cdot 10^6}{3.0\cdot 10^{8}} = 10^{-2} \frac{T}{m}$$
02

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: $$I = \frac{1}{2} \cdot 3.0\cdot 10^8 \cdot 8.854 \times 10^{-12} \cdot (3.0 \cdot 10^6)^2$$ $$I = 1.197 \times 10^{11} W/m^2$$
03

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.

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