Ch. 31 - Maxwell's Equations and Electromagnetic Waves

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 31 - Maxwell's Equations and Electromagnetic Waves

Problem 48

Chapter 30, Problem 48

A satellite beams microwave radiation with a power of 16 kW toward the Earth’s surface, 550 km away. When the beam strikes Earth, its circular diameter is about 1500 m. Find the rms electric field strength of the beam.

Verified step by step guidance

Understand the problem: We are tasked with finding the root mean square (rms) electric field strength of a microwave beam. The given data includes the power of the beam (P = 16 kW), the distance to the Earth's surface (550 km), and the diameter of the beam on Earth's surface (1500 m). The relationship between power, intensity, and the electric field strength will be used.

Step 1: Calculate the intensity of the beam. Intensity (I) is defined as the power (P) per unit area (A). The area of the circular beam can be calculated using the formula for the area of a circle: \( A = \pi r^2 \), where \( r \) is the radius of the beam. Here, \( r = \frac{1500}{2} \) meters. Substitute this into the formula for intensity: \( I = \frac{P}{A} = \frac{P}{\pi r^2} \).

Step 2: Relate the intensity to the rms electric field strength. The intensity of an electromagnetic wave is related to the rms electric field strength (E_rms) by the formula: \( I = \frac{1}{2} c \varepsilon_0 E_{\text{rms}}^2 \), where \( c \) is the speed of light in a vacuum (\( 3 \times 10^8 \ \text{m/s} \)) and \( \varepsilon_0 \) is the permittivity of free space (\( 8.85 \times 10^{-12} \ \text{F/m} \)). Rearrange this formula to solve for \( E_{\text{rms}} \): \( E_{\text{rms}} = \sqrt{\frac{2I}{c \varepsilon_0}} \).

Step 3: Substitute the expression for intensity (\( I = \frac{P}{\pi r^2} \)) into the formula for \( E_{\text{rms}} \). This gives: \( E_{\text{rms}} = \sqrt{\frac{2 \cdot \frac{P}{\pi r^2}}{c \varepsilon_0}} \).

Step 4: Plug in the known values: \( P = 16 \times 10^3 \ \text{W} \), \( r = 750 \ \text{m} \), \( c = 3 \times 10^8 \ \text{m/s} \), and \( \varepsilon_0 = 8.85 \times 10^{-12} \ \text{F/m} \). Simplify the expression to find \( E_{\text{rms}} \).

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Key Concepts

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

Power and Intensity

Power is the rate at which energy is transferred or converted, measured in watts (W). Intensity, on the other hand, is the power per unit area, typically expressed in watts per square meter (W/m²). In this context, the intensity of the microwave beam can be calculated by dividing the power of the beam by the area over which it is distributed when it reaches the Earth's surface.

Electric Field Strength

The electric field strength (E) is a measure of the force per unit charge experienced by a charged particle in an electric field. It is related to the intensity of an electromagnetic wave, where the root mean square (rms) electric field strength can be derived from the intensity using the formula E = √(2 * I / ε₀), where I is the intensity and ε₀ is the permittivity of free space.

Area of a Circle

The area of a circle is calculated using the formula A = πr², where r is the radius. In this problem, the diameter of the beam is given, allowing us to find the radius (half of the diameter) and subsequently calculate the area over which the microwave power is distributed. This area is essential for determining the intensity of the beam at the Earth's surface.

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