02Class 12 Physics

Electrostatic Potential and Capacitance formulas

Potential, capacitor and dielectric formulas for Electrostatic Potential and Capacitance. Unlike field, potential is a scalar, so it just adds.

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Potential

Potential of a point charge: $\displaystyle V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}$
Potential of a dipole: $\displaystyle V = \frac{1}{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2}$
Zero on the equatorial plane.
Field from potential: $\displaystyle E = -\frac{dV}{dr}$
Potential energy of two charges: $\displaystyle U = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r}$
Dipole energy in a uniform field: $\displaystyle U = -\vec p\cdot\vec E$

Capacitance: $\displaystyle C = \frac{Q}{V}$
Parallel plate capacitor: $\displaystyle C = \frac{\varepsilon_0 A}{d}$
With a dielectric filling the gap: $\displaystyle C = \frac{K\varepsilon_0 A}{d}$
Series combination: $\displaystyle \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$
Same charge on each.
Parallel combination: $\displaystyle C = C_1 + C_2$
Same voltage across each.
Energy stored: $\displaystyle U = \frac{Q^2}{2C} = \tfrac12 CV^2 = \tfrac12 QV$
Energy density: $\displaystyle u = \tfrac12\varepsilon_0 E^2$

Mixing the capacitor combination rules up with resistors. They're the other way round.
Assuming charge stays constant in every dielectric problem. It does if the battery is disconnected, while voltage stays constant if it's connected.
Thinking E = 0 means V = 0. Inside a conductor E is zero, while V is constant and not necessarily zero.

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