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Chapter 22: Problem 27

The electric field in a microwave traveling through air has amplitude $0.60 \mathrm{mV} / \mathrm{m}\( and frequency \)30 \mathrm{GHz}$. Find the amplitude and frequency of the magnetic field.

Short Answer

Expert verified
Answer: The amplitude of the magnetic field associated with the microwave is 2 * 10^(-12) T, and the frequency is 30 GHz.

Step by step solution

01

Write down the given information

The amplitude of the electric field (E) is 0.60 mV/m and the frequency (f) is 30 GHz.
02

Convert the units to SI units

We need to convert the given values to SI units, so we'll convert mV/m to V/m and GHz to Hz: E = 0.60 * 10^(-3) V/m = 6.0 * 10^(-4) V/m f = 30 * 10^9 Hz
03

Calculate the angular frequency (蠅)

The angular frequency (蠅) is related to the frequency (f) by the following formula: 蠅 = 2蟺f Now, we'll substitute the given frequency into the equation and solve for the angular frequency: 蠅 = 2蟺(30 * 10^9 Hz) = 60蟺 * 10^9 rad/s
04

Calculate the speed of light in the medium (c)

As the microwave is traveling through air, we can assume that the speed of light in the medium is the same as the speed of light in vacuum. Therefore, c = 3 * 10^8 m/s.
05

Calculate the amplitude of the magnetic field (B)

Now that we know the amplitude and angular frequency of the electric field, we can use the following formula to calculate the amplitude of the magnetic field (B): B = E/c Substitute the values of E and c into the equation: B = (6.0 * 10^(-4) V/m) / (3 * 10^8 m/s) = 2 * 10^(-12) T The amplitude of the magnetic field is 2 * 10^(-12) T.
06

Frequency of the magnetic field

In an electromagnetic wave, the frequency of the electric field is the same as the frequency of the magnetic field. Therefore, the frequency of the magnetic field is 30 GHz. The final answer is: the amplitude of the magnetic field is 2 * 10^(-12) T, and the frequency is 30 GHz.

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