You measure the capacitance of two capacitors with uncertainties, C1 = (1190 ± 25) μF and C2 = (842 ± 12) μF. You connect these capacitors in series. What is the uncertainty in the value for [math: 1C1] \frac{1}{C_1} ?
[math: Δ(1C1)=] \Delta (\frac{1}{C_1}) = Answer Question 8[input] 1/μF (3 sig. fig.) Note: this is asking for just the uncertainty, not the value, only enter the uncertainty into the box

Question

You measure the capacitance of two capacitors with uncertainties, C1 = (1190 ± 25) μF and C2 = (842 ± 12) μF. You connect these capacitors in series. What is the uncertainty in the value for [math: 1C1] \frac{1}{C_1} ?
[math: Δ(1C1)=] \Delta (\frac{1}{C_1}) =  Answer Question 8[input] 1/μF (3 sig. fig.) Note: this is asking for just the uncertainty, not the value, only enter the uncertainty into the box

BlackTom AI · Accepted Answer

To tackle this, I start by identifying the quantity whose uncertainty we need: Δ(1/C1). The relationship between a small change in C and the corresponding change in 1/C is given by the derivative: d(1/C) ≈ -dC / C^2, and for uncertainty magnitudes we use Δ(1/C) ≈ ΔC / C^2.

First, extract the given values: C1 = 1190 μF and its uncertainty ΔC1 = 25 μF. These will be plugged into the propagation formula.

Compute the square of the capacitance: C^2 = (1190 μF)^2. This equals 1190 × 1190 = 1,416,100 μF^2.

Now substitute into the uncertainty formula: Δ(1/C1) ≈ ΔC1 / C1^2 = 25 μF / 1,416,100 μF^2.

Carry out the division: 25 / 1,416,100 ≈ 1.765 × 10^-5. The units work out to 1/μF, since the μF units cancel appropriately with the μF^2 in the denominator.

For clarity, note that this approach uses a first-order (linear) error propagation, which is appropriate when the uncertainties are small relative to the measured value. Any higher-order terms are neglected in this approximation, which is standard practice for small ΔC compared to C.

If needed for reporting, you would round to the desired significant figures, keeping in mind the original uncertainty of ΔC1 (two significant figures).

PHYS1221-PHYS1231-Physics 1B T1 2025

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