prelecture_08_capacitors

Capacitors

Overview

• Parallel-Plate Capictor
  • Discuss what happens when we insert a dielectric
• Prallel and Series Combination of capacitors

Parallel-plate Capacitor

• Last time

  • [C \equiv \frac{ Q}{ \Delta V} = \frac{ Q}{ Ed} = \frac{ Q}{ \frac{ \sigma}{ \varepsilon_0} d} = \frac{ Q}{ \frac{ Q}{ \varepsilon_0 A} d} = \frac{ \varepsilon_0 A}{ d}]

• if [d] increases

  • [Q] remains constant
  • [E = \frac{ \sigma}{ \varepsilon_0}] remains constant
  • [\Delta V = - \int{ \vec E \cdot d \vec l} = E d] increases
  • [C = \frac{ Q}{ \Delta V} = \frac{ \varepsilon_0 A}{ d}] decreases
  • [U] increases
    • [W_{byField}] is negative
    • [\Delta U = - W_{byField}] is positve

Question 1

Two parallel plates have length [a] and width [b] and are separated by [a] distance [d] (which is much smaller than [a] and [b]). If all of the dimensions are doubled, how does the capacitance of the system change?

• The capacitance increases by a factor of 2
  • The capacitance of a parallel-plate system is proportional to the area of the plates divided by the distance between them, which in this case equals [\frac{ a b}{ d}]. If [a], [b] and [d] are all doubled this becomes [ \frac{ 2 a 2 b}{ 2d} = \frac{ 2 a b}{ d}], which is twice as big as before.