Question

A. A parallel-plate capacitor is made from two aluminum-foil sheets, each 5.5 cm wide and 5.8 m long. Between the sheets is a Teflon strip of the same width and length that is 3.2×10−2 mm thick. What is the capacitance of this capacitor? (The dielectric constant of Teflon is 2.1.) B. Suppose that after walking across a carpeted floor you reach for a doorknob and just before you touch it a spark jumps 0.55 cm from your finger to the knob. Find the minimum voltage needed between your finger and the doorknob to generate this spark. The minimum field that can produce a spark in air is 3.0×106V/m

Answer 1

Answer 2

Part A: Capacitance of the Parallel-Plate Capacitor

Given:

• Dimensions of each aluminum-foil sheet: Width = 5.5 cm, Length = 5.8 m

• Thickness of Teflon strip ($d$) = $3.2\times 10^{-2}$ mm = $3.2\times 10^{-5}$ m

• Dielectric constant of Teflon ($\kappa$) = 2.1

• Permittivity of free space (${ϵ}_0$) = $8.85\times 10^{-12}$ F/m

• 
  

Steps:

1. Calculate the area ($A$) of one plate:

   $$A=\text{width}\times \text{length}=5.5\text{cm}\times 5.8\text{m}=0.055\text{m}\times 5.8\text{m}=0.319{\text{m}}^2$$

2. Use the capacitance formula for a parallel-plate capacitor with a dielectric:

   $$C=\kappa {ϵ}_0\frac{A}{d}$$

   Plug in the values:

   $$C=2.1\times 8.85\times 10^{-12}\times \frac{0.319}{3.2\times 10^{-5}}$$$$C=2.1\times 8.85\times 10^{-12}\times 9,968.75$$$$C=2.1\times 8.82\times 10^{-8}$$$$C\approx 1.85\times 10^{-7}\text{F}=0.185\mu \text{F}$$

Answer for Part A:
The capacitance of the capacitor is approximately $\boxed{{\displaystyle 0.19\mu \text{F}}}$.

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Solution to Part B: Minimum Voltage to Generate a Spark

Given:

• Spark distance ($d$) = 0.55 cm = $5.5\times 10^{-3}$ m

• Minimum electric field ($E$) to produce a spark in air = $3.0\times 10^6$ V/m

• 
  

Steps:

1. The voltage ($V$) required is given by:

   $$V=E\times d$$

   Plug in the values:

   $$V=3.0\times 10^6\times 5.5\times 10^{-3}$$$$V=16,500\text{V}$$

Answer for Part B:
The minimum voltage needed is $\boxed{{\displaystyle 16,500\text{V}}}$.