Electric charge is conserved in all physical processes.

• Total electric charge in an isolated system remains constant
• Charge cannot be created or destroyed, only transferred
• $\sum q_{initial} = \sum q_{final}$

Examples:

• Rubbing a rod: charge moves between objects, total unchanged
• Beta decay: neutron -> proton + electron (charge conserved)
• Particle reactions: total charge before = total charge after

The electrostatic force between two point charges:

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

Vector form: $\vec{F}_{12} = k\frac{q_1 q_2}{r^2}\hat{r}_{21}$

Where:

• $k = 8.99 \times 10^9$ N - m2/C2 (Coulomb's constant)
• $q_1, q_2$ = charges (Coulombs)
• r = separation distance
• $\hat{r}_{21}$ = unit vector from q2 to q1

Properties:

• Inverse square law ($\propto 1/r^2$)
• Like charges repel, opposite charges attract
• Force acts along line connecting charges
• Superposition applies

The net force on a charge due to multiple charges is the vector sum of individual forces:

$\vec{F}_{net} = \sum_{i} \vec{F}_i = \sum_{i} k\frac{q Q_i}{r_i^2}\hat{r}_i$

Procedure for finding net force:

1. Calculate magnitude of each force using Coulomb's law
2. Determine direction of each force
3. Add as vectors (component method or head-to-tail)

Example: Force on charge q from q1 and q2: $\vec{F} = \vec{F}_1 + \vec{F}_2$ $F_x = F_1\cos\theta_1 + F_2\cos\theta_2$ $F_y = F_1\sin\theta_1 + F_2\sin\theta_2$

Electric field is the force per unit charge at a point in space:

$\vec{E} = \frac{\vec{F}}{q_{test}}$

Or equivalently: $\vec{E} = \lim_{q_{test} \to 0} \frac{\vec{F}}{q_{test}}$

Units: N/C or V/m

Physical meaning:

• Field exists independent of test charge
• Test charge must be small enough not to disturb source
• Tells us what force WOULD be experienced if a charge were placed there

Electric field produced by a single point charge:

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

Where:

• q = source charge
• r = distance from source charge
• $\hat{r}$ = unit vector pointing away from positive charge

Direction:

• Points away from positive charges
• Points toward negative charges
• Same as direction of force on positive test charge

Electric field lines are a visual representation of electric fields.

Properties:

• Begin on positive charges, end on negative charges
• Can also begin/end at infinity
• Never cross each other
• Density of lines ∝ field strength
• Tangent to field lines gives field direction

Patterns:

• Single positive charge: lines radiate outward
• Single negative charge: lines radiate inward
• Dipole (+ and -): lines from + to -
• Parallel plates: uniform field between plates

Uniform field has constant magnitude and direction throughout region.

Sources:

• Between large, closely-spaced parallel plates
• $E = V/d$ for parallel plates

Properties:

• Field lines are parallel and equally spaced
• $E = \text{constant}$
• Force on charge: $\vec{F} = q\vec{E}$ (uniform force)

Motion of charged particle in uniform E-field:

• Acceleration: $\vec{a} = \frac{q\vec{E}}{m}$
• Parabolic trajectory (like projectile motion)

when charge moves from A to B:

$W_{A \to B} = -\Delta U = U_A - U_B$

For constant field: $W = qE d$

For point charge field: $U = k\frac{q_1 q_2}{r}$

Key points:

• Conservative force (work independent of path)
• $\oint \vec{E} \cdot d\vec{l} = 0$ around closed loop
• Work equals negative change in potential energy

Electric potential energy of two point charges:

$U = k\frac{q_1 q_2}{r}$

Physical meaning:

• Work required to bring charges from infinity to separation r
• Positive: repulsive (like charges)
• Negative: attractive (opposite charges)

For system of multiple charges: $U_{total} = \sum_{i<j} k\frac{q_i q_j}{r_{ij}}$

Superposition principle applies for potential energy.

Electric potential V is potential energy per unit charge:

$V = \frac{U}{q}$

Units: Volts (V) = J/C

For point charge: $V = k\frac{q}{r}$

Key points:

• Scalar quantity (not a vector)
• Independent of test charge
• Potential of single charge set by convention: $V_\infty = 0$

Potential difference between two points:

$\Delta V = V_B - V_A = \frac{U_B - U_A}{q} = \frac{W_{A \to B}}{q}$

In terms of electric field: $\Delta V = -\int_A^B \vec{E} \cdot d\vec{l}$

For uniform field: $\Delta V = -E d \cos\theta$

Physical meaning:

• Voltage tells us how much energy gained/lost per charge
• Driving force in circuits
• 1 V = 1 J/C of energy per charge

Equipotential: all points at same electric potential.

Properties:

• No work required to move charge along equipotential
• $\vec{E}$ is perpendicular to equipotential surfaces
• Cannot have E-field component along equipotential

Relationship: $V = \text{constant along equipotential}$

Patterns:

• Point charge: equipotentials are spheres (concentric circles in 2D)
• Parallel plates: equipotentials are parallel planes
• Dipole: more complex pattern

Electric field is the negative gradient of potential:

$\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$

In one dimension: $E = -\frac{dV}{dx}$

For uniform field: $E = -\frac{\Delta V}{\Delta d}$

Physical meaning:

• E-field points in direction of decreasing potential
• E-field strength = rate of change of potential
• E-fields can be found from potential (scalar easier to work with)

In electrostatic equilibrium:

$\vec{E}_{inside} = 0$

Why:

• Free charges can move in conductor
• If E ≠ 0, charges would move until E = 0
• Excess charge distributes to make E = 0

Consequences:

• Entire conductor is at same potential
• No net force on charges inside
• Charge resides entirely on surface

Excess charge on conductor distributes to minimize potential.

Properties:

• All excess charge resides on surface
• Charge density varies with curvature (higher at sharp points)
• E-field at surface: $E = \frac{\sigma}{\varepsilon_0}$ (perpendicular to surface)

Hollow conductor:

• E = 0 inside cavity (if no charge inside)
• Charge on outer surface only
• Shielding: external fields don't affect interior

Applications:

• Faraday cage (electromagnetic shielding)
• Lightning rods (concentrate charge at tip)
• Electrostatic precipitators