Example 24.4 from College Physics, 24.4 Energy in Electromagnetic Waves

Calculate Microwave Intensities and Fields

On its highest power setting, a certain microwave oven projects 1.00 kW of microwaves onto a 30.0 by 40.0 cm area. (a) What is the intensity in ${W/m}^{2}$ ? (b) Calculate the peak electric field strength $E_{0}$ in these waves. (c) What is the peak magnetic field strength $B_{0}$ ?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
In part (a), we can find intensity from its definition as power per unit area. Once the intensity is known, we can use the equations below to find the field strengths asked for in parts (b) and (c).
Solution for (a)
Entering the given power into the definition of intensity, and noting the area is 0.300 by 0.400 m, yields

$I = \frac{P}{A} = \frac{1 . 00 kW}{0 . 300 m \times 0 . 400 m} .$
24.21

Here $I = I_{ave}$ , so that

$I_{ave} = \frac{1000 W}{0.120 m^{2}} = 8 . 33 \times 10^{3} {W/m}^{2} .$
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Note that the peak intensity is twice the average:

$I_{0} = 2 I_{ave} = 1 . 67 \times 10^{4} W / m^{2} .$
24.23
Solution for (b)
To find $E_{0}$ , we can rearrange the first equation given above for $I_{ave}$ to give

$E_{0} = {(\frac{2 I_{ave}}{{cε}_{0}})}^{1/2} .$
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Entering known values gives

$\begin{array}{l} E_{0} & = & \sqrt{\frac{2 (8.33 \times 10^{3} {W/m}^{2})}{(3.00 \times 10^{8} m/s) (8.85 \times 10^{- 12} C^{2} / N \cdot m^{2})}} \\ = & 2.51 \times 10^{3} V/m . \end{array}$
24.25
Solution for (c)
Perhaps the easiest way to find magnetic field strength, now that the electric field strength is known, is to use the relationship given by

$B_{0} = \frac{E_{0}}{c} .$
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Entering known values gives

$\begin{array}{l} B_{0} & = & \frac{2.51 \times 10^{3} V/m}{3.0 \times 10^{8} m/s} \\ = & 8.35 \times 10^{- 6} T . \end{array}$
24.27
Discussion
As before, a relatively strong electric field is accompanied by a relatively weak magnetic field in an electromagnetic wave, since $B = E / c$ , and $c$ is a large number.

0 of 5 steps open

How did it go?

Strategy

Solution for (a)

Solution for (b)

Solution for (c)

Discussion

Strategy

Solution for (a)

Solution for (b)

Solution for (c)

Discussion