Electric charge is a fundamental property of matter.

Two types of electric charge:

• Positive charge (protons)
• Negative charge (electrons)

Properties:

• Like charges repel each other
• Opposite charges attract
• Charge is quantized: $q = ne$ where e = 1.602 × 10-19 C
• Charge is conserved in all interactions

Materials:

• Conductors: charges free to move within material
• Insulators (dielectrics): charges bound to atoms, cannot move freely

The total electric charge in an isolated system remains constant.

$\sum q_{initial} = \sum q_{final}$

Implications:

• Charge cannot be created or destroyed
• Charge can only be transferred between objects
• Total charge of universe is constant

Examples:

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

Electrostatic force between two point charges:

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

Where:

• $k_e = 8.99 \times 10^9$ N - m2/C2 (Coulomb's constant)
• $q_1, q_2$ = charges in Coulombs
• r = separation distance in meters

Alternative form: $F = \frac{1}{4\pi\varepsilon_0} \frac{|q_1||q_2|}{r^2}$

Where $\varepsilon_0 = 8.85 \times 10^{-12}$ C2/(N - m2)

Complete vector expression for electrostatic force:

$\vec{F}_{12} = k_e \frac{q_1 q_2}{r^2}\hat{r}_{21}$

Where $\hat{r}_{21}$ is the unit vector from charge 2 to charge 1.

Direction:

• Force on q1 from q2:
• Along line connecting charges
• Toward q2 if opposite signs (attractive)
• Away from q2 if same signs (repulsive)

Sign convention:

• Positive result: repulsive (like charges)
• Negative result: attractive (opposite charges)

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

$\vec{F}_{net} = \sum_{i=1}^{N} \vec{F}_i$

For N charges: $\vec{F}_{on\,q} = \sum_{i=1}^{N} k_e \frac{q q_i}{r_i^2}\hat{r}_i$

Procedure:

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

Example: Force on charge q from q1 and q2: $\vec{F} = \vec{F}_1 + \vec{F}_2$ $F_x = F_{1x} + F_{2x}$ $F_y = F_{1y} + F_{2y}$

Point charge: $\vec{E} = k_e \frac{q}{r^2}\hat{r}$

Multiple point charges (superposition): $\vec{E} = \sum_{i=1}^{N} k_e \frac{q_i}{r_i^2}\hat{r}_i$

Continuous charge distribution (integral form): $\vec{E} = \int k_e \frac{dq}{r^2}\hat{r}$

Where dq depends on distribution type.

Common distributions:

• Line charge: $dq = \lambda dl$
• Surface charge: $dq = \sigma dA$
• Volume charge: $dq = \rho dV$

Electric field is the negative gradient of electric 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: $E = -\frac{dV}{dr}$

Physical meaning:

• E-field points in direction of decreasing potential
• E-field strength equals rate of potential change
• Potential difference: $\Delta V = -\int \vec{E} \cdot d\vec{l}$

For uniform field: $E = -\frac{\Delta V}{\Delta d} = \frac{V}{d}$ (between parallel plates)

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

$\vec{E} = \lim_{q_0 \to 0} \frac{\vec{F}}{q_0}$

Where:

• $\vec{F}$ = force experienced by test charge
• $q_0$ = test charge (must be small to not disturb field)

Units: N/C or V/m

Key points:

• Field exists independent of test charge
• Test charge must be positive (convention)
• Vector quantity (magnitude and direction)

For a single point charge q:

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

Where:

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

Direction:

• Away from positive charges
• Toward negative charges

Magnitude follows inverse square law: $E \propto 1/r^2$

Electric dipole: Two equal and opposite charges separated by small distance.

Dipole moment: $\vec{p} = q\vec{d}$

Where:

• q = magnitude of one charge
• $\vec{d}$ = vector from -q to +q

Torque on dipole in uniform E-field: $\vec{\tau} = \vec{p} \times \vec{E}$

Magnitude: $\tau = pE\sin\theta$

Force on dipole in non-uniform E-field: $\vec{F} = \nabla(\vec{p} \cdot \vec{E})$

Dipole aligns with field (minimum potential energy orientation)

Charge per unit length along a line.

$\lambda = \frac{dq}{dl}$

Units: C/m

For uniform line charge: $\lambda = \frac{Q}{L}$

Where Q = total charge, L = total length

Used for:

• Charged rods
• Charged lines
• Charged arcs

Charge per unit area on a surface.

$\sigma = \frac{dq}{dA}$

Units: C/m2

For uniform surface charge: $\sigma = \frac{Q}{A}$

Where A = total area

Used for:

• Charged plates
• Charged spheres
• Charged disks

Charge per unit volume.

$\rho = \frac{dq}{dV}$

Units: C/m3

For uniform volume charge: $\rho = \frac{Q}{V}$

Where V = total volume

Used for:

• Charged solid spheres
• Charged cylinders

E-field from finite uniformly charged line at perpendicular distance R:

$d\vec{E} = k_e \frac{\lambda dl}{r^2}\hat{r}$

Component analysis:

• Symmetry considerations simplify
• Use symmetry to cancel perpendicular components

General procedure:

1. Choose coordinate system (typically along line)
2. Express $dq = \lambda dx$
3. Express distance r in terms of x
4. Express $d\vec{E}$ components
5. Integrate appropriate components

Charged ring (on axis at distance x from center):

$E_x = \frac{k_e Q x}{(x^2 + R^2)^{3/2}}$

Where:

• Q = total charge
• R = ring radius
• x = distance along axis

Derivation:

• Symmetry: perpendicular components cancel
• Only axial component remains

Charged arc (at center):

• Depends on arc angle
• Use angular integration
• $\vec{E} = \int_0^\phi k_e \frac{\lambda R\,d\phi}{R^2}\hat{r}$

Electric field from infinite line charge at distance r:

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

Where $\lambda$ = linear charge density (C/m)

Direction:

• Radial from line (outward for + charge)
• Perpendicular to line

Properties:

• E ∝ 1/r (not 1/r2)
• Can also be found using Gauss's law
• Valid for truly infinite line

Electric field from infinite plane with surface charge density σ:

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

Units: N/C or V/m

Properties:

• Uniform and independent of distance!
• Perpendicular to plane
• Direction: away from + surface, toward - surface

Between two parallel plates (one +, one -): $E = \frac{\sigma}{\varepsilon_0}$

Where σ = charge density on each plate

Uniform electric field (between parallel plates):

Force on charge: $\vec{F} = q\vec{E}$

Acceleration: $\vec{a} = \frac{q\vec{E}}{m}$

Trajectory is parabolic (like projectile motion).

Equations of motion:

• Position: $\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2}\vec{a} t^2$
• Velocity: $\vec{v}(t) = \vec{v}_0 + \vec{a} t$

Deflection in parallel plate capacitor: $\Delta y = \frac{1}{2}at^2 = \frac{1}{2}\frac{qE}{m}\left(\frac{L}{v_x}\right)^2$

Where L = plate length, $v_x$ = horizontal velocity

Work done by electric force: $W_{A \to B} = \int_A^B \vec{F} \cdot d\vec{l} = q\int_A^B \vec{E} \cdot d\vec{l}$

Work-energy theorem: $W = \Delta K = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

For uniform field: $W = qEd\cos\theta = q\Delta V$

Applications:

• Particle accelerators
• Velocity determination after acceleration through potential difference
• Energy conservation in electrostatic systems