2401 Capacitors and Capacitance

  • Any two conductors separated by an insulator (or a vacuum) form a capacitor.
  • The ratio of charge to potential difference is called the capacitance of the capacitor.

C=QVab\begin{aligned} C = \frac{Q}{V_{ab}} \end{aligned}

CC - The Capacitance of a capacitor.
QQ - Magnitude of charge on each conductor.
VabV_{ab} - Potential difference between conductors (a has charge +Q+Q, and b has charge $-Q).

  • The SI unit of capacitance is called one farad (1F).

1F=1 farad=1C/V=1 coulomb/volt\begin{aligned} 1F = 1 \text{ farad} = 1 C/V = 1 \text{ coulomb/volt} \end{aligned}

Calculating Capacitance: Capacitors in Vacuum

Exercises

3, 7, 8, 9, 12

24.3 A parallel-plate air capacitor of capacitance 245 pF has a charge of magnitude 0.148μC0.148 \mu C on each plate. The plates are 0.328 mm apart. (a) What is the potential difference between the plates? (b) What is the area of each plate? (c) What is the electric-field magnitude between the plates? (d) What is the surface charge density on each plate?

Solution

The capacitance is C=2.45×10−10FC = 2.45 \times 10^{-10} F.
The magnitude of charge is Q=1.48×10−7CQ = 1.48 \times 10^{-7} C.
The distance between the plates is d=3.28×10−4md = 3.28 \times 10^{-4} m.
a. The potential difference is

Vab=QC=1.48×10−72.45×10−10=604.1VV _{ab}= \frac{Q}{C} = \frac{1.48 \times 10^{-7}}{2.45 \times 10^{-10}} = 604.1V

b. The area of each plate

C=ϵ0Ad⇒A=Cdϵ0=2.45×10−10×3.28×10−48.85×10−12=9.08×10−3m2\begin{aligned} C &= \epsilon_0\frac{A}{d} \\ \To A &= \frac{Cd}{\epsilon_0} = \frac{2.45 \times 10^{-10} \times 3.28 \times 10^{-4}}{8.85 \times 10^{-12}}\\ &= 9.08 \times 10^{-3} m^2 \end{aligned}

c. The electric field between the plates

E=Qϵ0A=1.48×10−78.85×10−12×9.08×10−3=1.84×106N/m\begin{aligned} E &= \frac{Q}{\epsilon_0 A} = \frac{1.48 \times 10^{-7}}{8.85 \times 10^{-12} \times 9.08 \times 10^{-3}}\\ &= 1.84 \times 10^6 N/m \end{aligned}

d. The surface charge density

λ=QA=1.48×10−79.08×10−3=1.63×10−5C/m2\begin{aligned} \lm &= \frac{Q}{A} = \frac{1.48 \times 10^{-7}}{9.08 \times 10^{-3}} = 1.63 \times 10^{-5} C/m^2 \end{aligned}