Physics 202: Lecture 5 & Physics 2B: Lecture 13

Resistance, Resistors, and Circuits

• Ohm’s law
• Electric Circuits
• Current
• Kirchhoff’s laws

Conductors in Electrostatic Equilibrium

• Excess charge resides on the surface of a conductor.
• The exterior electric field is perpendicular to the surface.
• The surface is an equipotential.
• The electric field inside is zero: \vec{E} = 0
• The interior is all at the same potential.
• Surface charge density and electric field strength are largest at sharp corners.

Capacitors

• When a circuit is closed and a voltage is applied, the capacitor quickly charges.
• The potential difference across the capacitor is the same as the battery: VB = 12 \text{ V}, VC = 12 \text{ V}
• Electrons effectively travel from one plate to the other.
• A capacitor can deliver energy faster than a battery but cannot hold as much energy as a similar-sized battery.

Supercapacitors

• Currently, most electric cars use lithium-ion batteries.
• Supercapacitors are being explored to power cars and already power some trains and buses.
• Advantages:
  • Can be charged or discharged very quickly (when braking, motors work in reverse to charge the capacitor).
  • Longer lifetime than a battery and safer.
• Disadvantages:
  • A plate (2D object) cannot hold as much energy as a battery (3D object).

Capacitance

• Definition: The ratio of the charge on one conductor to the potential difference between conductors.
• Equation: C = \frac{Q}{V}
• Units: Farad (F)
• The larger the capacitance, the more charge can be stored with a certain potential difference.
• Capacitors do not have a similar role as resistance.

Parallel Plate Capacitor

• Capacitance equation: C = \frac{\varepsilon_0 A}{d}
  • A is the area of the plate.
  • d is the distance of separation.
  • \varepsilon0 is the permittivity of free space: \varepsilon0 = 8.85 \times 10^{-12} \frac{C^2}{Nm^2}
• Plates with more area have a bigger capacitance.
• Closer plates have a bigger capacitance.
• Closer plates result in a larger electric field, leading to more charge.

Dielectrics in Capacitors

• Insulating sheets called dielectrics are sometimes placed between the plates of a capacitor.
• This allows the plates to be closer together, thus having a larger capacitance (without electrons flowing between).
• New relation for capacitance with a dielectric: C = \frac{\kappa \varepsilon_0 A}{d}
  • \kappa is the dielectric constant.
• When a dielectric is added, it polarizes, creating an electric field that partially cancels the original electric field.
• This allows the same charge to be held on the capacitor with a smaller potential, thus capacitance is increased.

Capacitors in Series and Parallel

• Series:
  • Equivalent Capacitance: \frac{1}{C{eq}} = \frac{1}{C1} + \frac{1}{C_2}
  • Charge is the same: Q{eq} = Q1 = Q_2
  • Voltage adds up: V{eq} = V1 + V_2
• Parallel:
  • Equivalent Capacitance: C{eq} = C1 + C_2
  • Charge adds up: Q{eq} = Q1 + Q_2
  • Voltage is the same: V{eq} = V1 = V_2
• Capacitors can be in both parallel and series within the same circuit.
• Two capacitors in parallel can be considered effectively as one capacitor.

Stored Energy in a Capacitor

• First electron: W = q \Delta V = q(0)
• Last electron: W = q \Delta V \approx qV
• Average: W = \frac{1}{2}qV
• Stored energy: U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C}
• When the plates are pulled a small distance further apart, the energy stored in the capacitor increases.