Physics

Physics

Problem

Let Δμ/μΔμ/μ and Δε/εΔε/ε denote small fractional changes in the permeability and permittivity of free space, respectively. Which relation correctly gives the corresponding fractional change Δc/cΔc/c in the speed of light? a) Δc/c=−(Δμ/μ+Δε/ε)\displaystyle \Delta c/c = -\bigl(\Delta \mu/\mu + \Delta \varepsilon/\varepsilon\bigr) b) Δc/c=Δμ/μ+Δε/ε2\displaystyle \Delta c/c = \dfrac{\Delta \mu/\mu + \Delta \varepsilon/\varepsilon}{2} c) Δc/c=−Δμ/μ+Δε/ε2\displaystyle \Delta c/c = -\dfrac{\Delta \mu/\mu + \Delta \varepsilon/\varepsilon}{2} d) Δc/c=−Δμ/μ−Δε/ε2\displaystyle \Delta c/c = -\dfrac{\Delta \mu/\mu - \Delta \varepsilon/\varepsilon}{2}
Intuitive explanation

The speed of light depends on the electromagnetic properties of the medium through c=1/μ0ϵ0c = 1/\sqrt{\mu_0 \epsilon_0}. When μ\mu and ϵ\epsilon change slightly, you need to use logarithmic differentiation to find how cc responds—this automatically handles the reciprocal square root structure. The key insight is that cc is inversely proportional to the square root of both parameters, so fractional changes combine with specific weights.

Intuitive explanation

The speed of light depends on the electromagnetic properties of the medium through c=1/μ0ϵ0c = 1/\sqrt{\mu_0 \epsilon_0}. When μ\mu and ϵ\epsilon change slightly, you need to use logarithmic differentiation to find how cc responds—this automatically handles the reciprocal square root structure. The key insight is that cc is inversely proportional to the square root of both parameters, so fractional changes combine with specific weights.

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