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(a) Consider the Galilean transformation along the \(x\)-direction : \(x' = x - vt\) and \(t' = t\). In frame \(S\) the wave equation for electromagnetic waves in a vacuum is $${\partial^2E(x, t)\over \partial x^2} - {1 \over c^2} {\partial^2E(x, t)\over \partial t^2} = 0$$ where \(E\) represents the electric field in the wave. Show that by using the Galilean transformation the wave equation in frame \(S'\) is found to be $$(1 - {v^2 \over c^2} ) {\partial^2E(x', t') \over \partial{x'^2}} + {2v \over c^2} {\partial^2E(x', t') \over \partial{x'} \partial{t'}} - {1 \over c^2} {\partial^2E(x', t') \over \partial{t'^2}} = 0$$ This has a different form than the wave equation in \(S\). Hence the Galilean transformation \(violates\) the first relativity postulate that all physical laws have the same form in all inertial reference frames. (\(Hint\): Express the derivatives \(\partial/\partial{x}\) and \(\partial/\partial{t}\) in terms of \(\partial/\partial{x'}\) and \(\partial/\partial{t'}\) by use of the chain rule.) (b) Repeat the analysis of part (a), but use the Lorentz coordinate transformations, Eqs. (37.21), and show that in frame \(S'\) the wave equation has the same form as in frame \(S\): $${\partial^2E(x', t') \over \partial{x'}^2} - {1 \over c^2} {\partial^2E(x', t') \over \partial{t'}^2} = 0$$ Explain why this shows that the speed of light in vacuum is c in both frames \(S\) and \(S'\).

Short Answer

Expert verified
The Galilean transformation alters the wave equation, violating relativity; the Lorentz transformation preserves its form, showing light speed is constant.

Step by step solution

01

Express Derivatives Using Galilean Transformation

The Galilean transformation is given by \(x' = x - vt\) and \(t' = t\). To express derivatives in terms of \(x'\) and \(t'\), use the chain rule.\[\frac{\partial}{\partial x} = \frac{\partial x'}{\partial x} \frac{\partial}{\partial x'} + \frac{\partial t'}{\partial x} \frac{\partial}{\partial t'} = \frac{\partial}{\partial x'},\]\[\frac{\partial}{\partial t} = \frac{\partial x'}{\partial t} \frac{\partial}{\partial x'} + \frac{\partial t'}{\partial t} \frac{\partial}{\partial t'} = -v \frac{\partial}{\partial x'} + \frac{\partial}{\partial t'}.\]
02

Apply the Chain Rule to Second Derivatives

Compute the second derivatives using the expressions obtained in Step 1:\[\frac{\partial^2}{\partial x^2} = \left(\frac{\partial}{\partial x'}\right)^2 = \frac{\partial^2}{\partial x'^2},\]\[\frac{\partial^2}{\partial t^2} = \left(\frac{\partial}{\partial t}\right)^2 = \left(-v \frac{\partial}{\partial x'} + \frac{\partial}{\partial t'}\right)^2 = v^2 \frac{\partial^2}{\partial x'^2} - 2v\frac{\partial^2}{\partial x' \partial t'} + \frac{\partial^2}{\partial t'^2}.\]
03

Substitute Back into Wave Equation

Substitute the expressions for the second derivatives into the wave equation:\[\frac{\partial^2 E}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 E}{\partial t^2} = \frac{\partial^2 E}{\partial x'^2} - \frac{1}{c^2}(v^2 \frac{\partial^2 E}{\partial x'^2} - 2v \frac{\partial^2 E}{\partial x' \partial t'} + \frac{\partial^2 E}{\partial t'^2})\]Simplify to obtain:\[(1 - \frac{v^2}{c^2}) \frac{\partial^2 E}{\partial x'^2} + \frac{2v}{c^2} \frac{\partial^2 E}{\partial x' \partial t'} - \frac{1}{c^2} \frac{\partial^2 E}{\partial t'^2} = 0.\]
04

Use Lorentz Transformation for Part (b)

The Lorentz transformation is given by:\[x' = \gamma (x - vt), \quad t' = \gamma (t - \frac{vx}{c^2}),\]where \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \). Use the chain rule to express derivatives in terms of \(x'\) and \(t'\):\[\frac{\partial}{\partial x} = \gamma \frac{\partial}{\partial x'} - \frac{v\gamma}{c^2} \frac{\partial}{\partial t'}, \quad \frac{\partial}{\partial t} = -v\gamma \frac{\partial}{\partial x'} + \gamma \frac{\partial}{\partial t'}.\]With these expressions, you can show that the wave equation retains its form:\[\frac{\partial^2 E}{\partial x'^2} - \frac{1}{c^2} \frac{\partial^2 E}{\partial t'^2} = 0.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Equation
The wave equation describes how waves, such as electromagnetic waves, propagate through space. In a vacuum, the wave equation for an electric field \( E(x, t) \) is given by:

\[ \frac{\partial^2 E(x, t)}{\partial x^2} - \frac{1}{c^2} \frac{\partial^2 E(x, t)}{\partial t^2} = 0 \]
This equation essentially states that the second spatial derivative of the electric field minus the second temporal derivative, adjusted by the speed of light squared, equals zero.

  • The term \( \frac{\partial^2}{\partial x^2} \) represents how the wave propagates in space.
  • The term \( \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \) represents how the wave changes over time.
  • The constant \( c \) is the speed of light, highlighting that electromagnetic waves travel at this consistent speed in a vacuum.
The Galilean transformation, which is a straightforward shift in coordinates between two frames, does not preserve the form of the wave equation due to its assumption that time is absolute. This results in discrepancies when considering the propagation of light, as examined in the exercise.
Lorentz Transformation
The Lorentz transformation is key to understanding how physical laws, including wave equations, retain their form across different inertial frames moving at constant velocities relative to each other. Unlike the Galilean transformation, which fails at relativistic speeds, the Lorentz transformation accounts for the constancy of the speed of light.

The transformation is described by the equations:
\[ x' = \gamma (x - vt), \quad t' = \gamma (t - \frac{vx}{c^2}), \]where \( \gamma \) is the Lorentz factor, defined as \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \).

  • The Lorentz transformation ensures that the speed of light \( c \) is the same in all reference frames.
  • It corrects for time dilation and length contraction, phenomena absent in Galilean physics.
  • Using this transformation on the wave equation demonstrates that the form remains unchanged in any inertial frame: \( \frac{\partial^2 E(x', t')}{\partial x'^2} - \frac{1}{c^2} \frac{\partial^2 E(x', t')}{\partial t'^2} = 0 \)
This invariance illustrates a fundamental principle of special relativity, ensuring that the laws of physics, including those describing wave propagation, are the same for observers in different inertial frames.
Electromagnetic Waves
Electromagnetic waves are oscillations of electric and magnetic fields that travel through space without the need for a medium. They are described by the wave equation, a crucial component in understanding their behavior.

These waves are part of the electromagnetic spectrum and include:
  • Radio waves
  • Microwaves
  • Infrared
  • Visible light
  • Ultraviolet
  • X-rays
  • Gamma rays
All electromagnetic waves propagate at the speed of light \( c \) in a vacuum, which is approximately 299,792 kilometers per second.

  • The electric and magnetic fields in these waves are perpendicular to each other and to the direction of propagation.
  • The constancy of the speed of electromagnetic waves in a vacuum was a core insight for Einstein's theory of relativity.
  • The wave equation reflects this constancy, emphasizing that the physics of electromagnetic waves remains consistent across different reference frames when using the Lorentz transformation.
Understanding electromagnetic waves is fundamental for exploring technologies and phenomena ranging from wifi and radio broadcasting to the behavior of light and radiation in scientific research.

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Most popular questions from this chapter

Inside a spaceship flying past the earth at three-fourths the speed of light, a pendulum is swinging. (a) If each swing takes 1.80 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control (on earth) who is watching the experiment? (b) If each swing takes 1.80 s as measured by a person at mission control, how long will it take as measured by the astronaut in the spaceship?

Electromagnetic radiation from a star is observed with an earth-based telescope. The star is moving away from the earth at a speed of 0.520c. If the radiation has a frequency of 8.64 \(\times\) 10\(^{14}\) Hz in the rest frame of the star, what is the frequency measured by an observer on earth?

Everyday Time Dilation. Two atomic clocks are carefully synchronized. One remains in New York, and the other is loaded on an airliner that travels at an average speed of 250 m/s and then returns to New York. When the plane returns, the elapsed time on the clock that stayed behind is 4.00 h. By how much will the readings of the two clocks differ, and which clock will show the shorter elapsed time? (\(Hint\): Since \(u \ll c\), you can simplify \(\sqrt{1 - u^2/c^2}\) by a binomial expansion.)

The positive muon (\(\mu^+\)), an unstable particle, lives on average 2.20 \(\times\) 10\(^{-6}\) s (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of 0.900c, what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?

What is the speed of a particle whose kinetic energy is equal to (a) its rest energy and (b) five times its rest energy?

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