The method and the results are on the capacitance lab. Do the analysis yourself first, then check the answers below.
Quick answer: Capacitance is the charge stored per unit potential difference, C = Q/ΔV, measured in farads. These answers go with the AP Physics C lab on capacitance: the graph, the calculations, the uncertainties and the conclusion explained, for all 7 analysis questions.
The answers use the sample data above. Your values will differ because paper thickness, pressure and air gaps vary, but 1/C against n should still be close to a straight line through the origin, and κ should come out between about 2 and 4.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:E&M Unit 10
Question 1 Linearise the data
Since C = κε₀A/(nt), 1/C = (t/κε₀A) n, which is proportional to n. The plotted points fall on a straight line close to the origin.
Question 2 Gradient
Gradient = (1.053 − 0.217)/4 = 0.209 nF⁻¹ per sheet = 2.09 × 10⁸ F⁻¹ per sheet.
Question 3 Dielectric constant
κ = t/(gradient × ε₀A) = (1.0 × 10⁻⁴)/[(2.09 × 10⁸)(1.77 × 10⁻¹³)] = 1.0 × 10⁻⁴/3.70 × 10⁻⁵, so κ ≈ 2.7.
Question 4 Air comparison
C(air) = ε₀A/t = (1.77 × 10⁻¹³)/(1.0 × 10⁻⁴) = 1.77 nF. The measured 4.6 nF is about 2.6 times larger, consistent with κ near 2.7.
Question 5 Intercept
The line should pass through the origin because 1/C → 0 as n → 0. A positive intercept in C would show stray capacitance or air gaps adding to the reading.
Question 6 Uncertainty
Fractional uncertainty in t is 10 % and in the gradient 3 %, so about 10.5 % in κ, giving κ = 2.7 ± 0.3.
Question 7 Reducing error
Pressing the stack under a heavier board squeezes out air pockets, and measuring 20 sheets at once cuts the thickness error. Both make t and C more trustworthy.
