In some experiments, very tiny distances or spaces ( ≈ nm ) can be measured by using capacitance. Consider forming an LC circuit using a parallel-plate capacitor with plate area A, and a known inductance L. When the plate separation is changed by ∆x, the circuit’s oscillation frequency will change by ∆f. Show that ∆x/x ≈ 2(∆f/f).

Verified step by step guidance

1. Start by recalling the formula for the capacitance of a parallel-plate capacitor: $C\; =\; (\epsilon ₀A)/x$, where $\epsilon ₀$ is the permittivity of free space, $A$ is the plate area, and $x$ is the plate separation.
2. The oscillation frequency of an LC circuit is given by $f\; =\; 1/(2\pi \surd (LC))$. Substitute the expression for $C$ into this formula to express $f$ in terms of $x$: $f\; =\; 1/(2\pi \surd (L(\epsilon ₀A)/x))$.
3. Simplify the expression for $f$: $f\; =\; 1/(2\pi )\surd (x/(L\epsilon ₀A))$. Notice that $f$ is proportional to the square root of $x$.
4. To find the relationship between changes in $x$ and $f$, take the derivative of $f$ with respect to $x$, or equivalently use the proportionality: $∆f/f\; =\; (1/2)(∆x/x)$.
5. Rearrange the proportionality to isolate $∆x/x$: $∆x/x\; \approx \; 2(∆f/f)$. This shows the desired relationship between the fractional changes in plate separation and oscillation frequency.

Key Concepts

Capacitance

Capacitance is the ability of a system to store electric charge per unit voltage. In a parallel-plate capacitor, it is determined by the area of the plates (A) and the distance between them (d), following the formula C = ε₀(A/d), where ε₀ is the permittivity of free space. Changes in plate separation directly affect capacitance, which in turn influences the oscillation frequency of an LC circuit.

LC Circuit

An LC circuit is an electrical circuit consisting of an inductor (L) and a capacitor (C) connected together. It can oscillate at a natural frequency determined by the values of L and C, given by the formula f = 1/(2π√(LC)). The frequency of oscillation is sensitive to changes in capacitance, which can occur due to variations in plate separation in a parallel-plate capacitor.

Frequency Change Relation

The relationship between changes in frequency (∆f) and changes in physical parameters (like plate separation ∆x) in an LC circuit can be derived from the dependence of capacitance on distance. The approximation ∆x/x ≈ 2(∆f/f) indicates that relative changes in plate separation are proportional to twice the relative changes in frequency, highlighting the sensitivity of the circuit's oscillation frequency to small physical alterations.