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32. Electromagnetic Waves

The Doppler Effect of Light

Physics

32. Electromagnetic Waves

The Doppler Effect of Light

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3:18 minutes

Problem 62

Textbook Question

A spaceship moving toward Earth at 0.65c transmits radio signals at 95.0 MHz. At what frequency should Earth receivers be tuned?

Verified step by step guidance

Identify the problem as a relativistic Doppler effect scenario, where the frequency of the signal received on Earth is affected by the relative motion of the spaceship and Earth.

Write the formula for the relativistic Doppler effect:

f r_{e} = \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}} \cdot f_{e}

, where

f_{r}

is the received frequency,

f_{e}

is the emitted frequency,

v

is the relative velocity of the source (spaceship) with respect to the observer (Earth), and

c

is the speed of light.

Substitute the given values into the formula:

f_{e} = 95.0 MHz

v = 0.65 c

. The formula becomes:

f_{r} = \sqrt{\frac{1 + 0.65}{1 - 0.65}} \cdot 95.0

Simplify the square root term:

\sqrt{\frac{1.65}{0.35}}

. This term represents the relativistic factor due to the motion of the spaceship.

Multiply the simplified relativistic factor by the emitted frequency

95.0 MHz

to find the received frequency

f_{r}

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Key Concepts

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

Doppler Effect

The Doppler Effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. In this scenario, as the spaceship approaches Earth, the frequency of the transmitted radio signals will appear higher to the Earth receivers due to the relative motion, which is a key aspect of the effect.

Relativistic Doppler Shift

The relativistic Doppler shift accounts for the effects of special relativity when dealing with objects moving at significant fractions of the speed of light. It modifies the classical Doppler effect equations to include time dilation, which is crucial for accurately calculating the observed frequency of signals from fast-moving sources like the spaceship.

Frequency and Wavelength Relationship

The relationship between frequency and wavelength is governed by the equation c = fλ, where c is the speed of light, f is the frequency, and λ is the wavelength. Understanding this relationship is essential for converting between frequency and wavelength when analyzing the signals received on Earth, especially when considering the effects of the Doppler shift.

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