Electric Fields

Overview

Electric fields describe the influence that electric charges have on the space around them. They provide a way to understand how charges interact at a distance.

Coulomb's Law

The force between two point charges:

$$F = k\frac{q_1 q_2}{r^2}$$

Where:

• $k = 8.99 \times 10^9$ N·m²/C² (Coulomb's constant)
• $k = \frac{1}{4\pi\varepsilon_0}$
• $\varepsilon_0 = 8.85 \times 10^{-12}$ C²/(N·m²) (permittivity of free space)
• $q_1, q_2$ = charges
• $r$ = distance between charges

Properties

• Like charges repel, opposite charges attract
• Force is along the line joining the charges
• Follows Newton's third law (equal and opposite)

Electric Field

Definition

Force per unit positive test charge:

$$\vec{E} = \frac{\vec{F}}{q_0}$$

Unit: N/C = V/m

Field Due to Point Charge

$$E = \frac{kq}{r^2}$$

Direction: radially outward for positive charge, inward for negative

Vector Form

$$\vec{E} = \frac{kq}{r^2} \hat{r}$$

Electric Field Lines

Properties

• Start on positive charges, end on negative charges
• Never cross
• Density indicates field strength
• Tangent gives field direction at each point

Patterns

• Point charge: radial lines
• Dipole: curved lines from + to −
• Parallel plates: uniform parallel lines

Superposition Principle

The total field from multiple charges:

$$\vec{E}_{\text{total}} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \cdots$$

Each field calculated independently, then added vectorially.

Electric Dipole

Two equal and opposite charges separated by distance $d$:

Dipole Moment

$$\vec{p} = q\vec{d}$$

Direction: from negative to positive charge

Field on Axis (far from dipole)

$$E = \frac{2kp}{r^3}$$

Field on Perpendicular Bisector

$$E = \frac{kp}{r^3}$$

Dipole in Uniform Field

Torque:

$$\tau = pE \sin(\theta)$$

Continuous Charge Distributions

Linear Charge Density

$$\lambda = \frac{dq}{dL} \quad \text{(C/m)}$$

Surface Charge Density

$$\sigma = \frac{dq}{dA} \quad \text{(C/m}^2\text{)}$$

Volume Charge Density

$$\rho = \frac{dq}{dV} \quad \text{(C/m}^3\text{)}$$

Field Calculation

$$\vec{E} = \int \frac{k \, dq}{r^2} \hat{r}$$

Common Field Configurations

Infinite Line Charge

$$E = \frac{\lambda}{2\pi\varepsilon_0 r} = \frac{2k\lambda}{r}$$

Infinite Plane (Sheet) of Charge

$$E = \frac{\sigma}{2\varepsilon_0}$$

Independent of distance!

Parallel Plate Capacitor

Between plates:

$$E = \frac{\sigma}{\varepsilon_0}$$

Examples

Example 1: Force Between Charges

Two 3 μC charges are 2 m apart. Find the force.

$$F = \frac{kq_1 q_2}{r^2} = \frac{8.99 \times 10^9 \times (3 \times 10^{-6})^2}{4}$$ $$F = 0.020 \text{ N (repulsive)}$$

Example 2: Field from Point Charge

Find field at 0.5 m from a 4 μC charge.

$$E = \frac{kq}{r^2} = \frac{8.99 \times 10^9 \times 4 \times 10^{-6}}{0.25}$$ $$E = 1.44 \times 10^5 \text{ N/C}$$

Example 3: Superposition

Two charges: +5 μC at $x = 0$ and −3 μC at $x = 4$ m. Find field at $x = 2$ m.

From +5 μC (pointing right):

$$E_1 = \frac{k(5 \times 10^{-6})}{4} = 1.12 \times 10^4 \text{ N/C}$$

From −3 μC (pointing right toward negative):

$$E_2 = \frac{k(3 \times 10^{-6})}{4} = 6.74 \times 10^3 \text{ N/C}$$

Total (both point right):

$$E = E_1 + E_2 = 1.79 \times 10^4 \text{ N/C (to the right)}$$

Example 4: Parallel Plates

Parallel plates have surface charge $\pm\sigma = 2 \times 10^{-6}$ C/m². Find field between them.

$$E = \frac{\sigma}{\varepsilon_0} = \frac{2 \times 10^{-6}}{8.85 \times 10^{-12}} = 2.26 \times 10^5 \text{ N/C}$$

Example 5: Motion in Electric Field

An electron enters a uniform field $E = 1000$ N/C. Find acceleration.

$$F = qE = 1.6 \times 10^{-19} \times 1000 = 1.6 \times 10^{-16} \text{ N}$$ $$a = \frac{F}{m} = \frac{1.6 \times 10^{-16}}{9.11 \times 10^{-31}} = 1.76 \times 10^{14} \text{ m/s}^2$$

Conductors in Electrostatic Equilibrium

• Electric field inside is zero
• Excess charge resides on surface
• Field just outside is perpendicular to surface
• Field at surface: $E = \sigma/\varepsilon_0$