(II) Starting from Eq. 36–16a, show that the Doppler shift in wavelength is, if v ≪ c ,

∆λ / λ = v/c .

Verified step by step guidance

1. Identify the original equation (Eq. 36-16a) which typically describes the Doppler shift for light from a moving source. This equation is usually given by \( f' = f \left( \frac{c \pm v}{c} \right) \), where \( f' \) is the observed frequency, \( f \) is the emitted frequency, \( c \) is the speed of light, and \( v \) is the velocity of the source relative to the observer.
2. Express the observed wavelength \( \lambda' \) in terms of the observed frequency \( f' \) using the relation \( \lambda' = \frac{c}{f'} \). Substitute for \( f' \) from the Doppler shift equation to get \( \lambda' = \frac{c}{f \left( \frac{c \pm v}{c} \right)} = \frac{c}{f} \cdot \frac{c}{c \pm v} = \lambda \frac{c}{c \pm v} \), where \( \lambda \) is the original wavelength.
3. Simplify the expression for \( \lambda' \) to find the change in wavelength \( \Delta \lambda = \lambda' - \lambda \). This gives \( \Delta \lambda = \lambda \frac{c}{c \pm v} - \lambda = \lambda \left( \frac{c}{c \pm v} - 1 \right) \).
4. Assuming \( v \ll c \), approximate \( \frac{c}{c \pm v} \) using a first-order Taylor expansion, \( \frac{1}{1 \pm x} \approx 1 \mp x \) for \( x \ll 1 \), which leads to \( \frac{c}{c \pm v} \approx 1 \mp \frac{v}{c} \). Substituting this back, we get \( \Delta \lambda \approx \lambda \left(1 \mp \frac{v}{c} - 1\right) = \mp \lambda \frac{v}{c} \).
5. Finally, find the fractional change in wavelength \( \frac{\Delta \lambda}{\lambda} \), which simplifies to \( \frac{\Delta \lambda}{\lambda} = \mp \frac{v}{c} \). For the case where the source is moving away, the minus sign applies, giving \( \frac{\Delta \lambda}{\lambda} = \frac{v}{c} \), which is the required Doppler shift in wavelength for \( v \ll c \).