Electrostatic Potential & Capacitance: Complete Mind Map & Formulas

Definition: Work done in moving a unit positive test charge from one point to another against electrostatic forces.
$$V_{AB} = V_B - V_A = \frac{W_{AB}}{q_0}$$

Key Properties: • It is a scalar quantity.
• SI Unit: Volt ($\text{V}$) or $\text{J C}^{-1}$ or $\text{N m C}^{-1}$.
• $1\text{ V} = 1\text{ J C}^{-1} = 1\text{ N m C}^{-1}$.

Definition: Work done in bringing a unit positive test charge from infinity ($\infty$) to a given point in an electric field without acceleration.
$$V = \frac{W_{\infty \to P}}{q_0}$$

Physical Significance: • Scalar quantity.
• Potential at a point is 1 volt if 1 joule of work is done in moving a positive charge of 1 coulomb from infinity to that point.

Formula & Symmetry: For a point charge $q$ at distance $r$: $$V = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}$$ • It is spherically symmetric.
• V varies inversely with distance: $V \propto \frac{1}{r}$.

General & Special Cases: • General Point ($\vec{r}, \theta$): $$V = \frac{1}{4\pi\varepsilon_0} \frac{\vec{p}\cdot\vec{r}}{r^3} = \frac{1}{4\pi\varepsilon_0} \frac{p\cos\theta}{r^2}$$ • Axial Line ($\theta = 0^\circ$ or $180^\circ$): $$V_{\text{axial}} = \pm \frac{1}{4\pi\varepsilon_0} \frac{p}{r^2}$$ • Equatorial Line ($\theta = 90^\circ$): $$V_{\text{equa}} = 0$$

Algebraic Sum Principle: By superimposition, total potential is the algebraic sum of potentials due to individual charges: $$V = \frac{1}{4\pi\varepsilon_0} \sum_{i=1}^{N} \frac{q_i}{r_i}$$ $$V = \frac{1}{4\pi\varepsilon_0} \left( \frac{q_1}{r_1} + \frac{q_2}{r_2} + \dots + \frac{q_N}{r_N} \right)$$

Potential Gradient relation: The rate of change of electric potential with respect to distance is called the potential gradient, which equals negative electric field intensity: $$E = -\frac{dV}{dr}$$ Electric field points in the direction in which potential decreases steepest.

Core Formulas & Analysis: • Electric potential at position vector $\vec{r}$ is given by line integral: $$V = -\int_{\infty}^{\vec{r}} \vec{E}\cdot d\vec{r}$$ • Differential relation (Potential Gradient): $$E = -\frac{dV}{dr}$$ • The direction of $\vec{E}$ is always along the direction of the steepest decrease of potential. SI unit of $\vec{E}$ can also be expressed as $\text{V m}^{-1}$.

Detailed Properties: Any surface that has same electric potential at every point is an equipotential surface.
(i) No work is done in moving a test charge over an equipotential surface ($W = 0$).
(ii) Electric field lines are always normal (perpendicular) to the equipotential surface.
(iii) Equipotential surfaces are close together in strong field regions and farther apart in weak field regions.
(iv) Two equipotential surfaces can never intersect each other.

System Energy Concepts: • Potential energy of a system of point charges is the total work done in assembling them from infinity to their respective locations.
• For two point charges $q_1$ and $q_2$ separated by distance $r_{12}$: $$U = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r_{12}}$$ • Electron Volt ($\text{eV}$): Energy gained by an electron when accelerated through a potential difference of 1 volt. $$1\text{ eV} = 1.6 \times 10^{-19}\text{ J}$$

System Energy Formulas: • For 3 point charges ($q_1, q_2, q_3$ with distances $r_{12}, r_{23}, r_{31}$): $$U = \frac{1}{4\pi\varepsilon_0} \left( \frac{q_1 q_2}{r_{12}} + \frac{q_2 q_3}{r_{23}} + \frac{q_3 q_1}{r_{31}} \right)$$ • For a general system of $N$ point charges (summing over all unique pairs $i < j$): $$U = \frac{1}{4\pi\varepsilon_0} \sum_{all\ pairs\ i < j} \frac{q_i q_j}{r_{ij}}$$

Work & Energy Equations: • Work done in turning a dipole from orientation $\theta_1$ to $\theta_2$: $$W = -pE(\cos\theta_2 - \cos\theta_1)$$ • Potential energy expression taking $\theta_1 = 90^\circ$ and $\theta_2 = \theta$: $$U = -\vec{p}\cdot\vec{E} = -pE\cos\theta$$ • Special Equilibrium States:
- Stable Equilibrium ($\theta = 0^\circ$): $U = -pE$ (Minimum energy)
- Unstable Equilibrium ($\theta = 180^\circ$): $U = +pE$ (Maximum energy)
- Zero Energy State ($\theta = 90^\circ$): $U = 0$

Material Classifications: • Conductors: Substances containing a large number of free electrons allowing large-scale charge movement when an external electric field is applied.
• Insulators (Dielectrics): Substances with negligible free charge carriers that restrict free electron flow, though internal charges can experience atomic polarization.
Key Electrostatic Properties of Conductors:
(i) Net electrostatic field inside a conductor is zero ($E = 0$).
(ii) Just outside the surface of a charged conductor, electric field is normal to the surface ($E = \frac{\sigma}{\varepsilon_0}\hat{n}$).

Key Principles & Properties: • Electrostatic Shielding: The phenomenon of protecting a region from any external electric field. It is based on the fact that the electric field inside a hollow charged conductor or cavity is always zero ($E = 0$).
• Additional Conductor Properties:
(iii) Net charge inside a conductor is zero; excess charge resides entirely on the outer surface.
(iv) Potential is constant within and on the surface of a conductor.
(v) Surface field is proportional to local surface charge density ($\sigma$).
• Capacitance of a Conductor: Defined as the charge required to raise its potential by a unit amount ($C = \frac{Q}{V}$).

Detailed Formulations: • Spherical Conductor: Capacitance is directly proportional to its radius $R$: $$C = 4\pi\varepsilon_0 R$$ • Capacitor: An arrangement of two conductors separated by an insulating (dielectric) medium used to store electrostatic charge and electrical energy.
• Capacitance of a Capacitor ($C = \frac{Q}{V}$): Charge required to be supplied to one conductor to increase potential difference between two conductors by unit amount.
• Farad (SI Unit): A capacitor has a capacitance of 1 farad if 1 coulomb of charge is transferred from one plate to another on applying a potential difference of 1 volt. $$1\text{ F} = \frac{1\text{ C}}{1\text{ V}}$$ Sub-units: $1\text{ mF} = 10^{-3}\text{ F}$, $1\text{ }\mu\text{F} = 10^{-6}\text{ F}$, $1\text{ pF} = 10^{-12}\text{ F}$.

Standard Geometry Formulation: It consists of two large parallel conducting plates, each of area $A$, separated by a small distance $d$ in vacuum/air. $$C = \frac{\varepsilon_0 A}{d}$$ Capacitance scales directly with plate surface area $A$ and inversely with separation distance $d$.

Special Geometries: • Spherical Capacitor: Consists of two concentric spherical conducting shells of inner radius $a$ and outer radius $b$: $$C = \frac{4\pi\varepsilon_0 ab}{b - a}$$ • Cylindrical Capacitor: Consists of two coaxial conducting cylinders of inner radius $a$, outer radius $b$, and common length $L$: $$C = \frac{2\pi\varepsilon_0 L}{\ln(b/a)} = \frac{2\pi\varepsilon_0 L}{2.303 \log_{10}(b/a)}$$

Network Rules: • Series Combination: Charge ($Q$) on each capacitor is identical, but potential difference across each varies inversely with capacitance. $$\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots$$ • Parallel Combination: Potential difference ($V$) across each capacitor is identical, while charge distributes proportional to capacitance ($Q \propto C$). $$C_p = C_1 + C_2 + C_3 + \dots$$

Comprehensive Energy Equations: • Energy Stored ($U$): The electrostatic work done in charging a capacitor of capacitance $C$ to potential $V$ with charge $q$: $$U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}$$ • Electrostatic Energy Density ($u$): Electrical energy stored per unit volume in an electric field region of intensity $E$: $$u = \frac{1}{2}\varepsilon_0 E^2$$

Formulations & Derivations: • Common Potential ($V$): When a number of conductors of capacitances $C_1, C_2, C_3 \dots$ at potentials $V_1, V_2, V_3 \dots$ carrying charges $q_1, q_2, q_3 \dots$ are connected together in contact, charge redistributes until a common potential is attained: $$V = \frac{\text{Total Charge}}{\text{Total Capacitance}} = \frac{q_1 + q_2 + q_3 + \dots}{C_1 + C_2 + C_3 + \dots} = \frac{C_1 V_1 + C_2 V_2 + C_3 V_3 + \dots}{C_1 + C_2 + C_3 + \dots}$$ • Loss of Energy ($\Delta U$): When two capacitors at different potentials are connected, electrostatic energy is lost (usually as heat/sparking): $$\Delta U = \frac{1}{2} \frac{C_1 C_2}{C_1 + C_2} (V_1 - V_2)^2$$

Comprehensive Dielectric Concepts: • Dielectric: An insulating substance that restricts free current flow but permits electrostatic forces through it via microscopic polarization.
• Polar vs Non-Polar Molecules:
- Non-Polar: Center of positive and negative charges coincide ($\text{H}_2, \text{O}_2, \text{CO}_2, \text{CH}_4$).
- Polar: Center of positive and negative charges do not coincide ($\text{HCl}, \text{NH}_3, \text{H}_2\text{O}$).
• Polarisation of Dielectric: When placed in an external electric field $E_0$, an induced internal field $E'$ opposes it, reducing the net field to $E = E_0 - E'$; capacitance consequently increases.
• Dielectric Constant ($K$): Ratio of capacitance with dielectric ($C$) to that in vacuum ($C_0$): $$K = \frac{C}{C_0} = \frac{E_0}{E_0 - E'} = \frac{E_0}{E}$$

Modified Capacitance Formulations: • Fully Filled Dielectric ($K$): $$C = K \cdot C_0 = \frac{K \varepsilon_0 A}{d}$$ • Partial Dielectric Slab of thickness $t$ ($t < d$): $$C = \frac{\varepsilon_0 A}{d - t + \frac{t}{K}}$$ • Partial Conducting Slab of thickness $t$ ($t < d$): Substituting $K \to \infty$ for a conductor yields: $$C = \frac{\varepsilon_0 A}{d - t}$$

Formulas with Medium Constant $K$: • Spherical Capacitor with Dielectric: $$C = 4\pi\varepsilon_0 K \left(\frac{ab}{b - a}\right)$$ • Cylindrical Capacitor with Dielectric: $$C = \frac{2\pi\varepsilon_0 K L}{\ln(b/a)} = \frac{2\pi\varepsilon_0 K L}{2.303 \log_{10}(b/a)}$$

Working Principle & Function: • Definition: An electrostatic generator capable of building up high potential differences of the order of $10^6\text{ volts}$ ($1\text{ MV}$).
• Core Principle: Based on the property that when a charged hollow conductor is brought into internal contact with a hollow conductor, it transfers the whole of its charge to the outer conductor irrespective of how high the potential of the latter may be.
• Key Action: Utilizes the action of sharp points (corona discharge) to continuously spray and transport charge to the large spherical dome terminal, used primarily for accelerating charged particles to high kinetic energies.