🔖 Topics

  • Definition of capacitance
  • Parallel plate capacitors
  • Dielectric materials
  • Electric fields around parallel plate capacitors

🎯 Objectives

  • Describe what capacitance is and how it relates to charge and voltage
  • Calculate the electric fields inside of a parallel plate capacitor
  • Describe the effect of a dielectric material on capacitance
  • Calculate the energy stored in a capacitor

📋 Sequence

  • Definition of capacitance: \( Q = CV \)
  • Energy stored in a capacitor: \( U = CV^2 / 2 \)
  • Derivation of \( C = A \kappa \epsilon_o / d \) for a parallel plate capacitor
  • Derivation of \( C = 4 \pi \epsilon_o R \) for a spherical conductor
  • Electric field around a parallel plate capacitor
  • Microscopic model of a capacitor and dielectric

🖥️ Animations, Simulations, Activities

Dielectric Breakdown With the Van de Graaff Generator

Capacitance is a function of geometry. For a conducting sphere (like the dome of the Van de Graaff generator), the capacitance is given by: \( C = 4 \pi \epsilon_o R \) where \( R \) is the radius of the sphere. If the dielectric breakdown of air is estimated to be 3 MV / m, estimate the charge stored on the Van de Graaff generator.

According to the Winsco docs, the diameter of the dome is 25 cm and can develop potentials up to 350,000 V.

https://winsco.com/product/van-de-graaff-generator/

📝 Practice Problems

  1. A parallel plate capacitor with a total plate area of \( 0.25 m^2 \) and a plate separation of 0.5 mm is connected to a 12 V battery. How much charge is stored on the capacitor?

  2. Suppose I insert a dielectric material into the capacitor in problem one with a dielectric constant of \( \kappa = 150 \). How much charge is stored on the capacitor?

✅ Partial Solutions

  1. 53.1 nC
  2. 7.97 mC

📘 Connected Resources

  • Giambattista, Alan, et al. College Physics With an Integrated Approach to Forces and Kinematics. 5th ed., McGraw-Hill Education, 2020.