Unit 2 - Electric Potential & Conductors, Capacitors, Dielectrics
Electric Potential
Potential for Point Charges
$V = \sum_{i=1}^{n} \frac{1}{4 \pi \epsilon_{0}} \, (\frac{Q_i}{r})$
- Sum up all electric potentials that act on a given point from any given ion.
Potential of Conducting Spheres
$\Delta V = V_a - V_b = - \int_{a}^{b} E \, dr$
For a singular point charge, $V = \frac{1}{4 \pi \epsilon_{0}} \, (\frac{Q}{r})$
- For conducting spheres, all charge is concentrated on the surface.
- As such, outside the sphere, the potential behaves like point charge.
Work Done by Electric Field
$W = - q \Delta V$
- Positive Charges: Electric field does negative work on a positive charge to go from high to low potential.
- $\Delta V$ is negative $ \rightarrow W$ is negative
- Negative Charges: Electric Field does negative work to move from low to high potential.
- $\Delta V$ is positive, while $q$ is negative $ \rightarrow W$ is negative
Work Done by Applied Forces
$W_{applied} = q \Delta V$
- Positive work is done to counteract and go against the electric field due to the applied force.
Capacitors & Dielectrics
Capacitance for Different Capacitors
- Cylindrical: $C_{0} = \frac{2 \pi \epsilon_{0} \ell}{\ln (\frac{b}{a})}$
- Spherical: $C_{0} = 4 \pi \epsilon_{0} \frac{(b \times a)}{b- a}$
- Parallel Plate: $C_{0} = \frac{\epsilon_{0} A}{d}$
Effect of Diaelectrics
Vaccum (No Dielectric)
For all capacitors:
For parallel plate capacitors:
- $|\vec{E}|$ $= \frac{\sigma}{\epsilon_{0}}$, where $\sigma = \frac{Q}{A}$
- $|\vec{E}|$ $= \frac{V}{d}$
Some Percentage of Diaelectric
- Capacitance is equal to the fraction of capacitance without any dialectric plus the fraction of capacitance with dialectric added together.
Completely Filled with Diaelectric
For all capacitors:
For parallel plate capacitors:
- $|\vec{E}|$ $= \frac{\sigma}{\epsilon}$, where $\epsilon = \kappa \epsilon_{0}$
Relationships regarding Parallel Plate Capacitors
Relationships regarding Parallel Plate Capacitors
| Connected to Battery |
Disconnected to Battery |
| $V = V_{0}$ |
$V = \frac{V_{0}}{\kappa}$ |
| $C = \kappa C_{0}$ |
$C = \kappa C_{0}$ |
| $Q = \kappa Q_{0}$, $\sigma = \kappa \sigma_{0}$ |
$Q = Q_{0}$, $\sigma = \sigma_{0}$ |
| $E = E_{0}$ |
$E = \frac{E_{0}}{\kappa}$ |
| $U = \kappa U_{0}$ |
$U = \frac{U_{0}}{\kappa}$ |
Capacitors in Series and Parallel
Series
$\frac{1}{C_{tot}} = \sum_{i=1}^{n} \frac{1}{C_i}$
Parallel
$C_{tot} = \sum_{i=1}^{n} C_i$
Energy Stored in a Capacitor
$U_C = \frac{Q^2}{2C} = \frac{1}{2} QV = \frac{1}{2} C \, (V)^2$