Electrostatic Potential, Dipoles, and Conductors - Comprehensive Study Notes

Conceptual Overview:
- Electrostatic Potential ( $V$ ) and Electrostatic Potential Energy ( $U$ ) are key concepts that allow the analysis of electrical systems using scalar values rather than vectors (like Electric Fields).
- The study follows the logic of Gravitational Potential Energy: moving a mass against gravity stores energy; similarly, moving a charge against electrical repulsion stores electrostatic potential energy.
Fundamental Definition of Potential Energy Change ( $\Delta U$ ):
- $\Delta U = U_f - U_i = -W_{\text{internal conservative}} = W_{\text{external agent}}$ .
- This holds true if the work is done slowly or without change in kinetic energy ( $\Delta K = 0$ ).
System of Two Point Charges:
- Potential energy of a system with charges $q_1$ and $q_2$ at separation $r$ : $U = \frac{k q_1 q_2}{r}$ .
- Charges must be substituted with their sign.
- Interaction occurs because of the work done to bring the second charge from infinity ( $U_{\infty} = 0$ ) to distance $r$ .
System of Multiple Charges:
- To find the total potential energy of a system, sum the interaction energy of every unique pair.
- For $n$ charges, the number of terms is calculated as $\binom{n}{r}$ or $n C_2$ .
- Example (Square with 4 charges): 6 terms (4 sides + 2 diagonals).

Definition: The electric potential at a point is the external work done per unit charge to bring a test charge from infinity to that point.
- $V = \frac{U}{q}$ .
- Unit: Volts ( $V$ ).
- Nature: It is a scalar quantity. Unlike electric fields, you simply add potentials arithmetically (not vector-wise).
Potential Due to a Point Charge: $V = \frac{k q}{r}$ .
Relation Between Electric Field ( $E$ ) and Potential ( $V$ ):
- $dV = -\mathbf{E} \cdot d\mathbf{r}$ .
- $\Delta V = V_2 - V_1 = -\int_{r_1}^{r_2} \mathbf{E} \cdot d\mathbf{r}$ .
- In a uniform electric field ( $E_0$ ), the potential decrease in the direction of the field over distance $d$ is $\Delta V = E_0 \times d$ .
Calculating Field from Potential:
- $E_x = -\frac{\partial V}{\partial x}$ , $E_y = -\frac{\partial V}{\partial y}$ , $E_z = -\frac{\partial V}{\partial z}$ .
- Electric Field points from high potential to low potential.

Ring:
- At the center ( $O$ ): $V = \frac{k Q}{R}$ .
- On the axis at distance $x$ : $V = \frac{k Q}{\sqrt{R^2 + x^2}}$ .
Arc: For any arc subtending an angle at the center with charge $Q$ : $V = \frac{k Q}{R}$ .
Disk: On the axis at distance $x$ from the center:
- $V = \frac{\sigma}{2 \epsilon_0} (\sqrt{R^2 + x^2} - x)$ .
Hollow Sphere (Conducting or Shell):
- Outside ( $r > R$ ): $V = \frac{k Q}{r}$ .
- Surface ( $r = R$ ): $V = \frac{k Q}{R}$ .
- Inside ( $r < R$ ): $V = \frac{k Q}{R}$ (Constant; same as surface).
Solid Non-Conducting Sphere (Uniformly Charged):
- Outside ( $r > R$ ): $V = \frac{k Q}{r}$ .
- Inside ( $r < R$ ): $V = \frac{k Q}{2 R^3} (3 R^2 - r^2)$ .
- Center ( $r = 0$ ): $V_{\text{center}} = 1.5 \times \frac{k Q}{R}$ (1.5 times the surface potential).

Dipole Moment ( $\mathbf{p}$ ): $\mathbf{p} = q \times \mathbf{d}$ . Directed from negative charge to positive charge.
Electric Field of a Short Dipole:
- Axial Point: $E_{\text{axial}} = \frac{2 k p}{r^3}$ . (Direction: Along $\mathbf{p}$ ).
- Equatorial Point: $E_{\text{equatorial}} = \frac{k p}{r^3}$ . (Direction: Opposite to $\mathbf{p}$ ).
- General Point ( $r, \theta$ ): $E = \frac{k p}{r^3} \sqrt{1 + 3 \cos^2(\theta)}$ .
Potential of a Short Dipole: $V = \frac{k p \cos(\theta)}{r^2}$ .
Dipole in a Uniform External Field ( $E$ ):
- Net Force: $F_{\text{net}} = 0$ .
- Torque: $\boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}$ or $\tau = p E \sin(\theta)$ .
- Potential Energy ( $U$ ): $U = -\mathbf{p} \cdot \mathbf{E} = -p E \cos(\theta)$ .
- Work Done to Rotate from $\theta_1$ to $\theta_2$ : $W_{\text{ext}} = p E (\cos(\theta_1) - \cos(\theta_2))$ .
- SHM: If slightly displaced, time period $T = 2\pi \sqrt{\frac{I}{p E}}$ , where $I$ is Moment of Inertia.

Core Properties:
- Net electric field inside the material of a conductor is always zero ( $E_{\text{in}} = 0$ ).
- Electric field just outside the surface: $E = \frac{\sigma}{\epsilon_0}$ , always perpendicular to the surface.
- The entire conductor is an equipotential body.
- Any net charge given to a conductor resides only on its outer surface.
Induction and Shielding:
- Placing a charge $+q$ in a cavity inside a conductor induces $-q$ on the inner surface and $+q$ on the outer surface (to maintain neutrality).
- Electrostatic shielding protects sensitive equipment from external fields as the field inside a conductor cavity remains zero if no charge is inside.
Earthing/Grounding:
- When a conductor is grounded, its potential becomes zero ( $V = 0$ ) by exchanging charge with the Earth.

Mechanical Energy Conservation: In problems where charges move, use $K_i + U_i = K_f + U_f$ . For systems of charges, $U = \sum \frac{k q_i q_j}{r_{ij}}$ .
64 Identical Drops Problem:
- When 64 small drops (each with charge $q$ , radius $r$ , potential $V$ ) combine to form a large drop:
- Volume is conserved: $\frac{4}{3} \pi R^3 = 64 \times \frac{4}{3} \pi r^3 \rightarrow R = 4r$ .
- Total charge $Q = 64q$ .
- New potential $V_{\text{new}} = \frac{k Q}{R} = \frac{k (64q)}{4r} = 16 V_{\text{old}}$ .
Parallel Plate Charge Distribution:
- For two parallel conducting plates with total charges $Q_1$ and $Q_2$ :
- Outermost surfaces carry equal charge: $Q_{\text{outer}} = \frac{Q_1 + Q_2}{2}$ .
- Facing inner surfaces carry equal and opposite charges.
Self Potential Energy ( $U_{\text{self}}$ ):
- Hollow Sphere: $U_{\text{self}} = \frac{k Q^2}{2R}$ .
- Solid Sphere: $U_{\text{self}} = \frac{3 k Q^2}{5R}$ .

Q: Can Electric Field be zero at a point where Potential is non-zero?
- A: Yes. Inside a charged hollow conductor, $E = 0$ , but $V = \frac{k Q}{R} \neq 0$ .
Q: How does a dipole behave in a non-uniform field?
- A: It experiences both a net force and a torque. Use $F = -\frac{dU}{dr}$ to find force, where $U = -\mathbf{p} \cdot \mathbf{E}$ .
Q: What is the significance of the 2024 JEE Advanced Wire Question?
- A: It required calculating the potential difference due to an infinite wire ( $V_A - V_B = 2k\lambda \ln(\frac{r_B}{r_A})$ ) and superimposing it with the potential of a shell.