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 46a

Chapter 31, Problem 46a

When the Voyager 2 spacecraft passed Neptune in 1989, it was 4.5×10⁹ km from the earth. Its radio transmitter, with which it sent back data and s, broadcast with a mere 21 W of power. Assuming that the transmitter broadcast equally in all directions, What signal intensity was received on the earth?

Verified step by step guidance

Step 1: Understand the concept of signal intensity. Signal intensity (I) is defined as the power (P) per unit area (A) over which the power is distributed. Mathematically, it is expressed as:

I = \frac{P}{A}

Step 2: Recognize that the transmitter broadcasts equally in all directions, forming a spherical wavefront. The surface area of a sphere is given by:

A = 4 π r^{2}

, where

r

is the distance from the source.

Step 3: Substitute the given values into the formula for the surface area of a sphere. The distance

r

4.5 \times 10^{9}

km, which should be converted to meters:

4.5 \times 10^{12}

m. Then calculate the area:

A = 4 π {4.5}^{\times 10^{12}}

Step 4: Use the formula for intensity:

I = \frac{P}{A}

. Substitute the power

P = 21

W and the calculated area

A

into the formula.

Step 5: Simplify the expression to find the intensity

I

. This will give the signal intensity received on Earth in units of W/m².

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Key Concepts

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

Inverse Square Law

The Inverse Square Law states that the intensity of a physical quantity, such as light or radio waves, decreases with the square of the distance from the source. This means that if you double the distance from the source, the intensity is reduced to one-fourth. In the context of the Voyager 2 spacecraft, this law helps calculate how the signal strength diminishes as it travels the vast distance to Earth.

Signal Intensity

Signal intensity refers to the power per unit area received from a transmitting source. It is typically measured in watts per square meter (W/m²). For the Voyager 2 scenario, understanding how to calculate the signal intensity at Earth involves knowing the power output of the transmitter and the distance it traveled, allowing us to determine how much of that power is received.

Power and Area Relationship

The relationship between power and area is crucial in understanding how energy spreads from a point source. When a transmitter emits power uniformly in all directions, the power is distributed over the surface area of a sphere, which increases with the square of the radius. This concept is essential for calculating the received signal intensity, as it involves dividing the total power by the surface area of the sphere at the distance of the receiver.

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What is the total energy density in an electromagnetic wave of intensity 1000 W/m²?

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At one instant, the electric and magnetic fields at one point of an electromagnetic wave are $\vec{E} = (200 \hat{i} + 300 \hat{j} - 50 \hat{k}) V/m$ and $\vec{B} = B_{0} (7.3 \hat{i} - 7.3 \hat{j} + a \hat{k}) μ T$ . What are the values of $a$ and $B sub 0$ ?

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