Starting from Eq. 44–3, show that the Doppler shift in wavelength is ∆λ/λᵣₑₛₜ ≈ v/c (Eq. 44–6) for v ≪ c. [Hint: Use the binomial expansion.]

Verified step by step guidance

1. Start with the relativistic Doppler shift formula for wavelength: $\backslash (\backslash lambda\backslash )\; =\; \backslash (\backslash lambda\backslash )\_\{\backslash (\backslash text\{rest\}\backslash )\}\; \backslash (\backslash sqrt\{\backslash frac\{1\; +\; v/c\}\{1\; -\; v/c\}\backslash )\}$, where $\backslash (\backslash lambda\backslash )$ is the observed wavelength, $\backslash (\backslash lambda\backslash )\_\{\backslash (\backslash text\{rest\}\backslash )\}$ is the wavelength at rest, $v$ is the relative velocity, and $c$ is the speed of light.
2. Express the change in wavelength as $\backslash (\backslash Delta\backslash )\; \backslash (\backslash lambda\backslash )\; =\; \backslash (\backslash lambda\backslash )\; -\; \backslash (\backslash lambda\backslash )\_\{\backslash (\backslash text\{rest\}\backslash )\}$. Dividing through by $\backslash (\backslash lambda\backslash )\_\{\backslash (\backslash text\{rest\}\backslash )\}$, we get $\backslash (\backslash frac\{\backslash Delta\; \backslash lambda\}\{\backslash lambda\_\{\backslash text\{rest\}\backslash )\}\}\; =\; \backslash (\backslash frac\{\backslash lambda\}\{\backslash lambda\_\{\backslash text\{rest\}\backslash )\}\}\; -\; 1$.
3. Substitute the expression for $\backslash (\backslash lambda\backslash )$ from the Doppler shift formula into $\backslash (\backslash frac\{\backslash lambda\}\{\backslash lambda\_\{\backslash text\{rest\}\backslash )\}\}$: $\backslash (\backslash frac\{\backslash lambda\}\{\backslash lambda\_\{\backslash text\{rest\}\backslash )\}\}\; =\; \backslash (\backslash sqrt\{\backslash frac\{1\; +\; v/c\}\{1\; -\; v/c\}\backslash )\}$.
4. For $v\; \backslash (\backslash ll\backslash )\; c$, use the binomial expansion approximation: $\backslash (\backslash sqrt\{1\; +\; x\}\backslash )\; \backslash (\backslash approx\backslash )\; 1\; +\; \backslash (\backslash frac\{x\}\{2\}\backslash )$ for small $x$. Apply this to both the numerator and denominator of $\backslash (\backslash sqrt\{\backslash frac\{1\; +\; v/c\}\{1\; -\; v/c\}\backslash )\}$, expanding to first order in $v/c$.
5. Simplify the resulting expression to show that $\backslash (\backslash frac\{\backslash Delta\; \backslash lambda\}\{\backslash lambda\_\{\backslash text\{rest\}\backslash )\}\}\; \backslash (\backslash approx\backslash )\; \backslash (\backslash frac\{v\}\{c\}\backslash )$, which is the desired result for small velocities relative to the speed of light.

Key Concepts

Doppler Effect

The Doppler Effect refers to the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. In the context of light, when a source moves away from an observer, the observed wavelength increases (redshift), while if it moves towards the observer, the wavelength decreases (blueshift). This effect is crucial for understanding how motion affects the perception of waves, particularly in astrophysics.

Wavelength Shift

Wavelength shift is the change in the wavelength of a wave due to relative motion between the source and the observer. In the case of the Doppler Effect, this shift can be quantified as ∆λ, which represents the difference between the observed wavelength and the rest wavelength (λᵣₑₛₜ). Understanding this shift is essential for analyzing how velocities affect wave properties, especially in scenarios where the source is moving at speeds much less than the speed of light (v ≪ c).

Binomial Expansion

The binomial expansion is a mathematical technique used to approximate expressions of the form (1 + x)ⁿ for small values of x. In the context of the Doppler shift, it allows for simplifying the relationship between the observed wavelength and the rest wavelength when the velocity of the source is much smaller than the speed of light. This approximation is key to deriving the relationship ∆λ/λᵣₑₛₜ ≈ v/c, making complex calculations more manageable.