Electrostatic Potential and Capacitance Study Notes

Definition of Electric Potential ( $V$ ): The electric potential at a point is defined as the work done ( $W$ ) in bringing a unit positive test charge from infinity to that point without any acceleration.
Mathematical Representation: The potential ( $V$ ) is given by the formula: $V = \frac{W}{q_0}$
Potential Difference: This refers to the difference in electric potential between two points, representing the work done in moving a unit charge from one point to another within an electric field.
Units of Measurement:
- The SI unit of electric potential is the Volt ( $V$ ).
- It can also be expressed as Joules per Coulomb ( $J/C$ ).
- Unit equivalence: $\text{1 Volt} = \frac{1\text{ Joule}}{1\text{ Coulomb}}$

Potential due to a Point Charge: For a charge $Q$ at a distance $r$ , the potential is: $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$
Potential due to a Charged Spherical Shell (Radius $R$ ):
- Outside the shell ( $r > R$ ): The potential at a distance $r$ from the center is: $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$
- On the surface ( $r = R$ ): The potential is: $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{R}$
- Inside the shell ( $r < R$ ): The potential remains constant and is equal to the potential on the surface: $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{R}$

Equipotential Surface: An equipotential surface is a surface where the electric potential remains constant at every point over the entire surface.
Work Done: Since the potential difference between any two points on an equipotential surface is zero, the work done in moving a charge along such a surface is zero ( $W = 0$ ).
Behavior of Conductors in an Electric Field:
- The electric field inside a conductor is zero ( $E = 0$ ).
- Electric field lines are always perpendicular to the surface of a conductor.
- Any excess charge on a conductor resides entirely on its outer surface.
Equilibrium: Mention is made of charges being initially in stable equilibrium.

Definition of Capacitance ( $C$ ): Capacitance is defined as the ratio of the magnitude of the charge ( $Q$ ) on either of the plates to the potential difference ( $V$ ) across the plates.
Fundamental Formula: $Q = C \times V$ $C = \frac{Q}{V}$
Unit: The unit of capacitance is the Farad ( $F$ ), which is equal to one Coulomb per Volt.
Geometric Dependence: The capacitance of a capacitor depends on the geometry of the plates (area $A$ and separation $d$ ).
Parallel Plate Capacitor Formula:
- In vacuum: $C_0 = \frac{\epsilon_0 A}{d}$
- With a dielectric medium: If a dielectric medium with constant $K$ is placed between the plates: $C = \frac{K\epsilon_0 A}{d}$ $C = K \times C_0$

Dielectric Constant ( $K$ ): When a dielectric is inserted, it results in a net electric field inside the dielectric that is smaller than the original field ( $E = E_0/K$ ). This occurs because the dielectric produces an internal electric field in the opposite direction to the applied field.
Induced Charge ( $q'$ ): The charge induced on the surfaces of the dielectric is given by the formula: $q' = q \times (1 - \frac{1}{K})$
Potential Changes: When a dielectric is introduced and the capacitor is disconnected from the battery, the potential $(V)$ decreases by a factor of $K$ , while the charge $(Q)$ remains constant. If the battery remains connected, the potential remains constant and the charge increases.

Series Combination:
- The charge ( $Q$ ) remains the same on each capacitor.
- The total potential difference ( $V$ ) is the sum of individual potential differences across each capacitor: $V = V_1 + V_2 + V_3$
- The equivalent capacitance ( $C_s$ ) is calculated as: $\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$
Parallel Combination:
- The potential difference ( $V$ ) remains the same across each capacitor.
- The total charge ( $Q$ ) is the sum of the charges on each capacitor: $Q = Q_1 + Q_2 + Q_3$
- The equivalent capacitance ( $C_p$ ) is the sum of individual capacitances: $C_p = C_1 + C_2 + C_3 + \dots$

Connecting Capacitors with Opposite Polarity: When the positive terminal of one capacitor ( $C_1$ , $V_1$ ) is connected with the negative terminal of another ( $C_2$ , $V_2$ ), the common potential ( $V_{12}$ ) is: $V_{12} = \frac{C_1 V_1 - C_2 V_2}{C_1 + C_2}$
Redistribution of Charge: After connection, the charges redistribute until a common potential is reached. The ratio of the final charges is equal to the ratio of their capacitances: $\frac{q_1}{q_2} = \frac{C_1}{C_2}$
Loss of Energy: During the process of connecting capacitors and the subsequent charge flow, energy is dissipated as heat in the connecting wires.
Formula for Energy Loss ( $\Delta U$ ): The loss of energy is defined as the difference between the initial energy and the final energy: $\Delta U = U_{initial} - U_{final}$ $\Delta U = \frac{C_1 C_2 (V_1 - V_2)^2}{2(C_1 + C_2)}$