Coulomb Law Electric Field Potential Capacitance Ohm Law Magnetic Force Lorentz Force

# Electricity and Magnetism: Fields, Circuits, and Charged-Particle Motion

## 1. Charge and interaction

Electric charge is a property of matter measured in coulombs. Protons carry positive elementary charge, electrons carry equal-magnitude negative charge, and neutrons are neutral. Like charges repel and unlike charges attract. In an isolated system total charge is conserved. Charge is also quantized: ordinary objects gain or lose whole electrons, so net charge occurs in multiples of the elementary charge e.

Conductors contain charges that can move through the material. Insulators bind charge more tightly. Charging by friction transfers electrons; charging by contact shares charge; induction rearranges charge without requiring direct contact.

## 2. Coulomb's law

Two point charges separated by distance r exert equal and opposite forces. The magnitude is:

Here k is approximately 8.99 × 10⁹ N·m²/C² in vacuum. The force points along the line joining the charges. Its inverse-square dependence means doubling separation reduces force to one quarter.

For several charges, use superposition: calculate the force vector from each source separately and add components. Never add magnitudes when directions differ.

## 3. Electric field

An electric field assigns a force-per-charge vector to every point:

The test charge is imagined small enough not to disturb the sources. A positive test charge accelerates along the field; a negative charge accelerates opposite it. The field of a point source charge is:

Field lines begin on positive charge and end on negative charge. Their tangent gives field direction and their density suggests strength. Lines never cross because the field at one point cannot have two directions.

## 4. Electric flux and Gauss's law

Electric flux measures field passing through a surface. For a uniform field through a flat area:

Gauss's law relates total flux through any closed surface to enclosed charge:

It is especially useful for spherical, cylindrical, or planar symmetry. In electrostatic equilibrium, the field inside a conductor is zero, excess charge resides on its surface, and the conductor is an equipotential.

## 5. Electric potential and energy

Electric potential energy belongs to a charge configuration. For two point charges:

Electric potential is potential energy per unit test charge. A point charge creates:

Potential is a scalar, so contributions add algebraically. One volt is one joule per coulomb. The change in potential energy for charge q moving through potential difference ΔV is:

The electric field points toward decreasing potential:

Equipotential lines are perpendicular to field lines. Moving along an equipotential requires no electric work.

## 6. Capacitors

A capacitor stores separated charge and electric-field energy. Capacitance is charge magnitude per potential difference:

For large parallel plates of area A separated by d in vacuum:

A dielectric increases capacitance by reducing the effective internal field. Energy stored in a charged capacitor has equivalent forms:

Capacitors in parallel share voltage and add directly. Capacitors in series carry equal charge and add reciprocally.

## 7. Current and resistance

Electric current is the rate at which charge crosses a surface:

Conventional current points as positive charge would move, opposite electron drift in a metal. For an ohmic component, potential difference is proportional to current:

Resistance depends on material resistivity ρ, length L, and cross-sectional area A:

Electrical power transferred by a component is:

Use the form matching known quantities. A resistor converts organized electrical energy mainly into thermal energy.

## 8. Series and parallel networks

Series components carry the same current. Their equivalent resistance is:

Parallel components share the same voltage. Their conductances add:

A parallel equivalent must be smaller than the smallest branch resistance. Real batteries can be modelled as an ideal emf in series with internal resistance, explaining why terminal voltage falls under load.

## 9. Kirchhoff's laws

Kirchhoff's current law expresses charge conservation at a junction:

Kirchhoff's voltage law expresses energy conservation around a closed loop:

Choose current directions freely. A negative solved current means the actual direction is opposite the guess. While traversing a resistor with current, potential changes by −IR; across an ideal source from negative to positive terminal, it rises by the emf.

## 10. RC time dependence

A resistor limits how quickly a capacitor charges. For a source voltage V₀, the capacitor voltage rises as:

During discharge:

The time constant τ = RC sets the scale. After one time constant a charging capacitor reaches about 63 percent of its final voltage, while a discharging capacitor retains about 37 percent.

## 11. Magnetic fields

Moving charge and magnetic materials create magnetic fields. Field direction is visualized with a compass or the right-hand rule. Around a long straight wire carrying current I:

The field circles the wire. A long ideal solenoid with turn density n produces an approximately uniform interior field:

Unlike electric field lines, magnetic field lines form closed loops; isolated magnetic monopoles have not been observed.

## 12. Lorentz force

The combined electric and magnetic force on a particle is:

The magnetic part has magnitude qvB sin θ and is perpendicular to both velocity and field. It therefore changes direction but does no work on a point charge. Electric fields can change speed and kinetic energy.

For velocity perpendicular to a uniform magnetic field, magnetic force supplies centripetal force:

Parallel velocity remains unchanged, producing a helix. The rotation direction reverses with charge sign.

## 13. Force on currents and magnetic torque

A straight wire segment in a magnetic field experiences:

A current loop has magnetic dipole moment μ = NIA and torque:

This is the operating idea behind simple motors and analogue meters. The corresponding potential energy is lowest when the dipole aligns with the field.

## 14. Electromagnetic induction

Magnetic flux through a loop is ΦB. Faraday's law says changing flux produces emf:

The minus sign is Lenz's law: induced current opposes the flux change that created it. Generators convert mechanical work to electrical energy by changing flux. Transformers use changing current in one coil to induce voltage in another.

## 15. Measurement and safety

An ammeter is placed in series and should have very low resistance. A voltmeter is placed in parallel and should have very high resistance. Never connect an ammeter directly across a source. Discharge capacitors safely before handling a circuit, and remember that high voltage is dangerous chiefly because it can drive harmful current through the body.

## 16. Field and potential reasoning

Potential is often easier to calculate because it is scalar. Once V is known, its spatial slope gives E. Near positive charge, potential is positive and decreases outward; near negative charge it is negative and rises toward zero outward. A point may have zero potential but a nonzero field because scalar cancellation does not imply vector cancellation.

Dimensional checks are valuable. Electric field can use N/C or V/m. Capacitance uses farads, resistance ohms, magnetic field teslas, and flux webers.

## 17. Python field and trajectory simulator

This program calculates electric potential and field from point charges, draws equipotentials and streamlines, then integrates a test particle in uniform electric and magnetic fields. SI-like scaled values keep the pictures readable.

import numpy as np
import matplotlib.pyplot as plt

k = 8.9875517923e9

def point_charge_fields(charges, xlim=(-2, 2), ylim=(-2, 2), points=181):
    x = np.linspace(*xlim, points)
    y = np.linspace(*ylim, points)
    X, Y = np.meshgrid(x, y)
    V = np.zeros_like(X)
    Ex = np.zeros_like(X)
    Ey = np.zeros_like(X)
    for q, xq, yq in charges:
        dx, dy = X - xq, Y - yq
        r2 = dx * dx + dy * dy
        safe = np.maximum(r2, 2e-3)
        r = np.sqrt(safe)
        V += k * q / r
        Ex += k * q * dx / (safe * r)
        Ey += k * q * dy / (safe * r)
    return X, Y, V, Ex, Ey

charges = [(2e-9, -0.7, 0.0), (-2e-9, 0.7, 0.0)]
X, Y, V, Ex, Ey = point_charge_fields(charges)

fig, ax = plt.subplots(figsize=(7, 6))
levels = np.linspace(-80, 80, 17)
ax.contour(X, Y, V, levels=levels, colors="gray", linewidths=0.7)
strength = np.hypot(Ex, Ey)
ax.streamplot(X, Y, Ex, Ey, color=np.log10(strength + 1),
              density=1.5, cmap="viridis")
for q, xq, yq in charges:
    ax.scatter(xq, yq, s=100, c="red" if q > 0 else "blue")
ax.set(xlabel="x", ylabel="y", title="Field lines and equipotentials")
ax.set_aspect("equal")
plt.tight_layout()

def boris_push(position, velocity, q_over_m, E, B, dt, steps):
    """Stable charged-particle update for uniform E and B."""
    path = np.empty((steps, 3))
    for i in range(steps):
        path[i] = position
        v_minus = velocity + q_over_m * E * dt / 2
        t = q_over_m * B * dt / 2
        s = 2 * t / (1 + np.dot(t, t))
        v_prime = v_minus + np.cross(v_minus, t)
        v_plus = v_minus + np.cross(v_prime, s)
        velocity = v_plus + q_over_m * E * dt / 2
        position = position + velocity * dt
    return path

path = boris_push(
    position=np.array([0.0, 0.0, 0.0]),
    velocity=np.array([1.0, 0.3, 0.7]),
    q_over_m=1.0,
    E=np.array([0.12, 0.0, 0.0]),
    B=np.array([0.0, 0.0, 1.0]),
    dt=0.01,
    steps=3000,
)

fig = plt.figure(figsize=(7, 5))
ax3 = fig.add_subplot(111, projection="3d")
ax3.plot(path[:, 0], path[:, 1], path[:, 2])
ax3.set(xlabel="x", ylabel="y", zlabel="z", title="Trajectory in combined E and B")
plt.tight_layout()
plt.show()

With E set to zero, the perpendicular kinetic speed should remain nearly constant. Reverse q/m to see the orbit reverse. Add an electric field parallel to B to create acceleration along the helix. Change the charge arrangement to a dipole, equal like charges, or a quadrupole and predict the symmetry before plotting.

## 18. Worked circuit example

Suppose 6 Ω and 3 Ω resistors are in parallel across 12 V. The equivalent resistance is 2 Ω, so the source current is 6 A. Branch currents are 2 A and 4 A, adding to 6 A at the junction. The source provides 72 W. The branches dissipate 24 W and 48 W, confirming energy conservation.

## 19. Worked selector example

Crossed electric and magnetic fields can select particles of one speed. If E and v × B forces oppose, undeflected particles satisfy qE = qvB. Their speed is therefore E/B, independent of charge and mass. Particles with other speeds bend. A subsequent magnetic-only region can separate mass-to-charge ratios.

## 20. Common mistakes

• Electric field is force per positive test charge, not the path a released charge must follow.
• Potential is scalar; field is vector.
• Magnetic force is perpendicular to velocity and cannot directly change kinetic energy.
• Current is not consumed by a resistor; energy is transferred.
• Parallel branches share voltage, while series components share current.
• A negative answer usually reports direction relative to an assumption, not an impossible result.

## 21. Final perspective

Electricity and magnetism become coherent when organized around charge, field, energy, and conservation. Coulomb's law builds electrostatics. Potential turns vector work into a scalar landscape. Circuit laws are charge and energy conservation in network form. Magnetic fields steer moving charge, while changing magnetic flux creates electric effects. These ideas join naturally in Maxwell's theory and electromagnetic waves.

## 22. Laboratory investigations

A useful electrostatics investigation maps potential on conductive paper. Apply a small voltage to shaped electrodes, record points with equal voltmeter readings, and draw equipotential curves. Electric-field arrows cross those curves at right angles and point toward lower potential. Closely spaced equipotentials indicate a stronger field. This experiment measures voltage directly and infers field from its spatial change, which is usually more practical than measuring force on a tiny test charge.

For circuits, construct several resistor networks and measure branch currents and voltage drops. Compare the measured equivalent resistance with predictions and include meter uncertainty and resistor tolerance. Plot voltage against current for a resistor, filament lamp, and diode. A straight line through the origin identifies constant resistance; curvature shows that resistance changes with temperature or operating point.

A magnetic-force investigation can send current through a wire segment resting in a known field. Reverse current or field independently and record the force direction. Vary current while holding geometry fixed; force should be proportional to current. Rotate the wire to verify the sine dependence. Keep wires within safe current ratings because resistive heating grows as current squared.

## 23. Materials and microscopic current

In a metal, an applied field gives mobile electrons a small average drift velocity superimposed on rapid random motion. Signal propagation through a circuit is much faster than electron drift because the electric field establishes throughout the conductor. A microscopic current relation is:

Here n is mobile-carrier number density, q is carrier charge magnitude, A is conductor area, and v_d is drift speed. This explains why a thicker wire has lower resistance: it provides more parallel paths for carriers.

Semiconductors have carrier densities that can be controlled by temperature, light, doping, and electric fields. Electrons and positively described holes both contribute to conduction. Diodes use junction fields to favour current in one direction, while transistors use a small control signal to regulate a larger current. Their behaviour is not generally ohmic, so V = IR with a single constant R is not a complete device model.

## 24. Additional quantitative checks

For an electron accelerated from rest through potential magnitude V, electrical potential energy becomes kinetic energy:

This classical result is reliable while speed remains well below the speed of light. In a uniform electric field between plates, a charged particle has constant transverse acceleration qE/m, analogous to projectile motion. In a perpendicular magnetic field it follows a circular arc instead. Comparing these paths makes the different energy roles of E and B visible.

For a parallel-plate capacitor, increasing plate area increases stored charge at fixed voltage. Increasing separation lowers capacitance, although mechanical work done separating isolated charged plates changes the energy differently than separation performed while a battery holds voltage fixed. Always state what remains connected and what quantity stays constant.

A sensible final answer includes units and limiting behaviour. Coulomb force should approach zero at large separation. Equivalent parallel resistance should fall as another branch is added. A magnetic orbit radius should grow with momentum and shrink with field strength. Such checks often reveal an inverted fraction or lost sign before any instrument does.