Electric Potential

The electrostatic force is conservative, which allows the introduction of a scalar electric potential field.

1. Path Independence and Conservative Nature

The work done by the electrostatic force on a test charge $q_0$ moving from point $A$ to point $B$ in any static electric field is strictly independent of the physical path taken: $$W_{A \to B} = \int_A^B \vec{F} \cdot d\vec{r} = q_0 \int_A^B \vec{E} \cdot d\vec{r}$$ Consequently, the line integral of the electrostatic field around any closed loop vanishes identically: $$\oint_C \vec{E} \cdot d\vec{r} = 0$$ By Stokes' theorem: $$\oint_C \vec{E} \cdot d\vec{r} = \int_S (\nabla \times \vec{E}) \cdot d\vec{A} = 0 \implies \nabla \times \vec{E} = 0$$ The static electric field is strictly **irrotational** (conservative).

2. Definition of Electric Potential ($V$)

The electric potential difference $\Delta V = V_B - V_A$ between points $A$ and $B$ is defined as the external work required per unit positive test charge to transport it from $A$ to $B$ at constant kinetic energy: $$\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{r}$$ Taking the standard reference point at infinity ($V(\infty) = 0$), the absolute electric potential at point $P$ is: $$V(\vec{r}) = -\int_\infty^{\vec{r}} \vec{E} \cdot d\vec{r}'$$ SI Unit: **Volt (V)**: $$1 \text{ Volt} = 1 \text{ Joule/Coulomb (J/C)}$$ Dimensionally: $[V] = M L^2 T^{-3} I^{-1}$.

3. Electric Potential Energy ($U$)

The electrostatic potential energy $U$ of a charge $q$ located at a point of electric potential $V$ is: $$U = q V$$ The electron-volt (eV) is defined as the kinetic energy acquired by an electron accelerated through a potential difference of 1 Volt: $$1 \text{ eV} = 1.602176634 \times 10^{-19} \text{ Joules}$$

Calculating the scalar electric potential is algebraically much simpler than computing the vector electric field directly.

1. Potential of an Isolated Point Charge

Integrating the radial field $\vec{E} = \frac{q}{4\pi\epsilon_0 r^2}\hat{r}$ from $\infty$ to $r$: $$V(r) = -\int_\infty^r \frac{q}{4\pi\epsilon_0 r'^2} dr' = \left[ \frac{q}{4\pi\epsilon_0 r'} \right]_\infty^r = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$$ Notice $V \propto 1/r$ (whereas field $E \propto 1/r^2$). By scalar superposition, the potential due to $N$ discrete point charges is: $$V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^N \frac{q_i}{|\vec{r} - \vec{r}_i|}$$

2. Potential of an Electric Dipole

Consider a dipole consisting of $+q$ at $(0, 0, a)$ and $-q$ at $(0, 0, -a)$ with dipole moment $p = 2qa$ aligned along the z-axis. At field point $(r, \theta)$ where distance $r \gg a$: $$r_+ \approx r - a \cos\theta, \quad r_- \approx r + a \cos\theta$$ $$V(r, \theta) = \frac{q}{4\pi\epsilon_0} \left( \frac{1}{r_+} - \frac{1}{r_-} \right) \approx \frac{q}{4\pi\epsilon_0} \left( \frac{r_- - r_+}{r^2} \right) = \frac{q (2a \cos\theta)}{4\pi\epsilon_0 r^2}$$ $$V(r, \theta) = \frac{1}{4\pi\epsilon_0} \frac{\vec{p} \cdot \hat{r}}{r^2} = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2}$$ Key properties of the dipole potential:

• Decays as $1/r^2$ (faster than the $1/r$ point charge monopole potential).
• Vanishes identically in the equatorial plane ($\theta = 90^\circ \implies \cos(90^\circ) = 0$).

3. Continuous Charge Distributions

For continuous sources: $$V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_{V'} \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|} dV'$$ $$V_{\text{surface}} = \frac{1}{4\pi\epsilon_0} \int_{S'} \frac{\sigma(\vec{r}')}{|\vec{r} - \vec{r}'|} dA', \quad V_{\text{line}} = \frac{1}{4\pi\epsilon_0} \int_{L'} \frac{\lambda(\vec{r}')}{|\vec{r} - \vec{r}'|} dl'$$

Because the electrostatic field is conservative ($\nabla \times \vec{E} = 0$), vector calculus guarantees that it can be expressed as the negative gradient of a scalar potential.

1. The Gradient Relation

Consider an infinitesimal displacement $d\vec{r} = dx \hat{i} + dy \hat{j} + dz \hat{k}$: $$dV = -\vec{E} \cdot d\vec{r} = - (E_x dx + E_y dy + E_z dz)$$ By the multivariable chain rule: $$dV = \frac{\partial V}{\partial x} dx + \frac{\partial V}{\partial y} dy + \frac{\partial V}{\partial z} dz$$ Equating coefficients: $$E_x = -\frac{\partial V}{\partial x}, \quad E_y = -\frac{\partial V}{\partial y}, \quad E_z = -\frac{\partial V}{\partial z}$$ In compact vector notation: $$\vec{E} = -\nabla V$$ where $\nabla = \hat{i} \frac{\partial}{\partial x} + \hat{j} \frac{\partial}{\partial y} + \hat{k} \frac{\partial}{\partial z}$ is the vector del operator.

2. Physical Interpretation

• The mathematical gradient $\nabla V$ points in the direction of maximum spatial increase of potential.
• Therefore, the electric field vector $\vec{E} = -\nabla V$ **always points in the direction of steepest potential decrease**.
• Positive charges naturally accelerate from regions of high electric potential toward regions of low electric potential.

3. Equipotential Surfaces and Orthogonality

An **equipotential surface** is the spatial locus of all points having identical electric potential ($V(x, y, z) = \text{constant}$). For any displacement $d\vec{r}$ lying entirely within an equipotential surface: $$dV = -\vec{E} \cdot d\vec{r} = 0$$ Because $d\vec{r}$ is non-zero, this requires: $$\vec{E} \perp d\vec{r}$$ Fundamental Geometric Theorem: The electric field vector is everywhere perpendicular (orthogonal) to equipotential surfaces. Equipotential lines and electric field lines form a mutually orthogonal coordinate mesh.

In an electrical conductor, valence electrons are free to migrate throughout the crystalline lattice. In static equilibrium, charge migration ceases, establishing several fundamental physical theorems.

1. Zero Internal Electric Field

Inside the bulk material of a conductor in electrostatic equilibrium: $$\vec{E}_{\text{internal}} = 0$$ *Proof:* If $\vec{E}$ were non-zero inside, the free electrons would experience forces $\vec{F} = -e\vec{E}$ and accelerate, generating macroscopic currents, contradicting the premise of static equilibrium.

2. Zero Net Internal Charge Density

Applying Gauss's law $\nabla \cdot \vec{E} = \rho / \epsilon_0$ to any interior region: $$\vec{E} = 0 \implies \rho = 0$$ *Theorem:* Any net excess electric charge deposited on an insulated conductor resides entirely on its outer geometric surface.

3. The Entire Conductor is an Equipotential Volume

For any two interior points $A$ and $B$: $$V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{r} = 0 \implies V_A = V_B = \text{constant}$$ The surface and entire interior of a conductor exist at the exact same scalar potential.

4. Electric Field Immediately Outside a Charged Conductor

Construct a Gaussian pillbox of area $dA$ straddling the conductor surface: The bottom face inside the metal has $\vec{E} = 0$. The curved side has zero flux. The top face outside has field $\vec{E} \perp$ surface: $$\oint \vec{E} \cdot d\vec{A} = E dA = \frac{dq}{\epsilon_0} = \frac{\sigma dA}{\epsilon_0} \implies E = \frac{\sigma}{\epsilon_0}$$ $$\vec{E} = \frac{\sigma}{\epsilon_0} \hat{n}$$ Notice this field is exactly double the field of a single non-conducting sheet ($\sigma / 2\epsilon_0$), because charge inside the conductor rearranges to produce zero field internally and constructive reinforcement externally.

5. Electrostatic Shielding (Faraday Cage)

In a hollow conducting shell with a cavity containing zero charge, $\vec{E} = 0$ everywhere inside the cavity, regardless of external electric fields. This is **electrostatic shielding**.

Robert J. Van de Graaff (1929) developed the electrostatic accelerator generator, capable of generating potentials exceeding 5 to 20 million Volts.

1. Working Principle

The Van de Graaff generator exploits two electrostatic principles:

1. Corona Discharge (Action of Sharp Points): At sharp conducting needles of radius of curvature $r$, the surface charge density $\sigma \propto 1/r$ becomes immense. When local field exceeds the dielectric breakdown strength of air ($E_{\text{breakdown}} \approx 3 \times 10^6 \text{ V/m}$), air molecules ionize, spraying ions onto an insulating moving belt.
2. Cavity Charge Transfer: When a charged conductor contacts the *interior* surface of a hollow conducting sphere, all charge transfers completely to the *exterior* surface, regardless of how high the sphere's potential already is!

2. Mathematical Limit on Terminal Potential

For a spherical high-voltage dome of radius $R$ carrying charge $Q$: $$V = \frac{1}{4\pi\epsilon_0} \frac{Q}{R}, \quad E = \frac{1}{4\pi\epsilon_0} \frac{Q}{R^2} = \frac{V}{R}$$ Hence, the maximum sustainable potential is limited by dielectric breakdown of the surrounding gas: $$V_{\max} = R \cdot E_{\text{breakdown}}$$ In atmospheric air ($E_b = 3 \text{ MV/m}$), a dome of radius $R = 1.0\text{ m}$ attains $V_{\max} = 3.0\text{ MV}$. Immersing the generator in high-pressure insulating sulfur hexafluoride ($SF_6$) gas raises $E_b$ fivefold, enabling tandem accelerators to reach 25 million Volts for nuclear transmutation experiments.