Ch 31: Electromagnetic Fields and Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 31: Electromagnetic Fields and Waves

Problem 18

Chapter 31, Problem 18

A radio receiver can detect signals with electric field amplitudes as small as 300 μV/m. What is the intensity of the smallest detectable signal?

Verified step by step guidance

Understand the relationship between the electric field amplitude and intensity. The intensity $ I $ of an electromagnetic wave is related to the electric field amplitude $ E $ by the formula: $ I = \frac{1}{2} \varepsilon_0 c E^2 $, where $ \varepsilon_0 $ is the permittivity of free space ($ 8.85 \times 10^{-12} \ \text{F/m} $) and $ c $ is the speed of light in a vacuum ($ 3.00 \times 10^8 \ \text{m/s} $).

Convert the given electric field amplitude $ E $ from microvolts per meter ($ \mu \text{V/m} $) to volts per meter ($ \text{V/m} $). Since $ 1 \ \mu \text{V/m} = 10^{-6} \ \text{V/m} $, the given value of $ 300 \ \mu \text{V/m} $ becomes $ 300 \times 10^{-6} \ \text{V/m} $.

Substitute the values of $ \varepsilon_0 $, $ c $, and the converted $ E $ into the intensity formula: $ I = \frac{1}{2} \varepsilon_0 c E^2 $.

Simplify the expression by squaring the electric field amplitude $ E $ and multiplying it with $ \varepsilon_0 $ and $ c $. Then divide the result by 2 to find the intensity.

Ensure the units are consistent throughout the calculation, and the final intensity $ I $ will be in watts per square meter ($ \text{W/m}^2 $).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Field Amplitude

Electric field amplitude refers to the maximum strength of an electric field, measured in volts per meter (V/m). In the context of radio signals, it indicates how strong the electric field component of the electromagnetic wave is. A higher amplitude means a stronger signal, which is easier for a receiver to detect.

Intensity of a Signal

The intensity of a signal is defined as the power per unit area carried by the wave, typically measured in watts per square meter (W/m²). It is proportional to the square of the electric field amplitude. Understanding this relationship is crucial for calculating the intensity from the given electric field amplitude.

Relationship Between Electric Field and Intensity

The intensity (I) of an electromagnetic wave is related to the electric field amplitude (E) by the equation I = (E²)/(Z), where Z is the impedance of free space (approximately 377 ohms). This relationship allows us to convert the electric field amplitude of the detected signal into its corresponding intensity, which is essential for understanding the signal's detectability.

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Textbook Question

Unpolarized light with intensity 350 W/m² passes first through a polarizing filter with its axis vertical, then through a second polarizing filter. It emerges from the second filter with intensity 131 W/m². What is the angle from vertical of the axis of the second polarizing filter?

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Textbook Question

The electric field of an electromagnetic wave in a vacuum is $E_{y} = (20.0 V/m) \cos [(6.28 \times 10^{8}) x - ω t]$ , where x is in m and t is in s. What are the wave's magnetic field amplitude?

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Textbook Question

FIGURE EX31.23 shows a horizontally polarized radio wave of frequency 1.0×10⁶ Hz traveling into the figure. The maximum electric field strength is 1000 V/m. What is the maximum magnetic field strength?

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Textbook Question

Show that the quantity ϵ₀(dΦ_e/dt) has units of current.

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Textbook Question

A microwave beam with a wavelength of 1.5 cm has an intensity of 25 W/m². What is the magnetic field amplitude?

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Textbook Question

The magnetic field of an electromagnetic wave in a vacuum is $B_{z} = (3.0 μ T) \sin [(1.00 \times 10^{7}) x - ω t]$ , where x is in m and t is in s. What are the wave's wavelength?

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