Electric Charge & Electric Field

Properties of Electric Charge

Electric charge is a fundamental property of matter, existing in two types: positive and negative. The behavior of charges underlies all electrostatic phenomena.

• Dual Charge Nature: Positive charge (e.g., glass rod rubbed with silk) and negative charge (e.g., plastic rod rubbed with fur).

• Force Rules: Like charges repel; opposite charges attract.

• Atomic Structure: Protons (+) and neutrons are in the nucleus; electrons (−) are mobile in the outer cloud. Protons are stationary in solids.

• Ions: Atoms with net charge. Positive ions (cations) result from electron loss; negative ions (anions) from electron gain.

• Quantization of Charge: Charge exists in integer multiples of the elementary charge $e \approx 1.602 \times 10^{-19}\ \mathrm{C}$: $q = ne$

• Conservation of Charge: The total charge in a closed system remains constant.

Materials & Charge Transfer

• Conductors: Materials (e.g., copper) where charges move freely.

• Insulators: Materials (e.g., nylon) where charges are bound and do not move freely.

• Conduction: Transfer of charge by direct contact; transferred charge has the same sign as the source.

• Induction: Charging without contact; leaves the conductor with a net charge opposite to the inducing rod.

• Four-Step Induction Process:

  1. Polarize: Bring charged rod near uncharged object to separate charges.

  2. Ground: Touch object to allow repelled charges to leave.

  3. Disconnect: Remove ground while rod is still near.

  4. Remove Rod: Remaining charges redistribute, leaving net charge.

• Polarization: In conductors, electrons shift macroscopically; in insulators, molecules rotate or align with the field (molecular polarization).

Coulomb’s Law & Superposition

• Coulomb’s Law: The force between two point charges $q_1$ and $q_2$ separated by distance $r$ is: $F = \frac{1}{4\pi\epsilon_0} \frac{|q_1 q_2|}{r^2}$ where $k = \frac{1}{4\pi\epsilon_0} \approx 9.0 \times 10^9\ \mathrm{N \cdot m^2/C^2}$.

• Action-Reaction: Forces between charges are equal in magnitude and opposite in direction (Newton’s Third Law).

• Superposition Principle: The net force on a charge is the vector sum of all individual forces: $\vec{F}_{\text{net}} = \sum \vec{F}_i$

The Electric Field (E)

• Definition: The electric field at a point is the force per unit positive test charge: $\vec{E} = \frac{\vec{F}'}{q'}$ (Units: N/C or V/m)

• Direction: The field points in the direction of the force on a positive test charge.

• Force on a Charge: $\vec{F} = q \vec{E}$

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

• Field Lines: Originate on positive charges and terminate on negative charges; denser lines indicate stronger fields. Uniform fields (e.g., between parallel plates) have equally spaced, parallel lines.

Vector Mathematics (Review)

• Component Notation: $\vec{A} = A_x \hat{i} + A_y \hat{j}$

• Magnitude: $A = \sqrt{A_x^2 + A_y^2}$

• Angle: $\theta = \arctan\left(\frac{A_y}{A_x}\right)$

• Resolving Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$

Example:

Three charges at the corners of a triangle: Use vector addition to find the net force on one charge by resolving each force into components and summing them.

Electric Potential & Capacitance

Potential Energy & Work

Electrostatic forces are conservative, meaning the work done is independent of the path taken.

• Work Done by Field: $W_{a \to b} = -\Delta U_E = U_a - U_b$

• Uniform Field: For a charge $q'$ moving a distance $s$ in a constant field $E$: $W_{a \to b} = q' E s$

• Charge Movement:

  • With $\vec{E}$: Positive charge loses potential energy; negative charge gains potential energy.

  • Against $\vec{E}$: Positive charge gains potential energy; negative charge loses potential energy.

Electric Potential & Voltage

• Definition: Electric potential is potential energy per unit charge: $V = \frac{U_E}{q'}$ (Units: Volt (V), $1\ \mathrm{V} = 1\ \mathrm{J/C}$)

• Potential Difference: $V_{ab} = V_a - V_b$; equals the work per unit charge done by the electric force.

• Point Charge Potential: At distance $r$ from charge $q$: $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$

• Uniform Field (Parallel Plates): $V_{ab} = E d$

Equipotential Surfaces

• Definition: Surfaces where the electric potential is constant everywhere.

• Properties:

  • Always perpendicular to electric field lines.

  • Never cross each other.

  • Conductors in equilibrium are equipotential volumes; inside, $E = 0$.

Capacitance & Parallel Plates

• Definition: Capacitance is the ability to store charge per unit potential difference: $C = \frac{Q}{V_{ab}}$ (Units: Farad (F), $1\ \mathrm{F} = 1\ \mathrm{C/V}$)

• Parallel-Plate Capacitor: For plate area $A$ and separation $d$: $C = \epsilon_0 \frac{A}{d}$ ($\epsilon_0 \approx 8.85 \times 10^{-12}\ \mathrm{F/m}$)

• Stored Energy: $U = \frac{1}{2} Q V = \frac{1}{2} C V^2 = \frac{1}{2} \frac{Q^2}{C}$

Capacitor Combinations

• Series:

  • Same charge on all capacitors ($Q_1 = Q_2 = Q$).

  • Voltages add: $V_{ab} = V_1 + V_2$.

  • Equivalent capacitance: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2}$ $C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$

• Parallel:

  • Charges add: $Q = Q_1 + Q_2$.

  • Same voltage across all: $V_1 = V_2 = V$.

  • Equivalent capacitance: $C_{eq} = C_1 + C_2$

Dielectrics

• Definition: Insulating material placed between capacitor plates.

• Dielectric Constant (K): $K \geq 1$; increases capacitance and reduces field/voltage for fixed charge.

• Effects:

  • Field: $E = E_0 / K$

  • Voltage: $V = V_0 / K$

  • Capacitance: $C = K C_0$

Current, Resistance, & Direct-Current Circuits

Electric Current & Drift Motion

• Current Definition: Net rate of flow of charge through a cross-section: $I = \frac{\Delta Q}{\Delta t}$ (Units: Ampere (A), $1\ \mathrm{A} = 1\ \mathrm{C/s}$)

• Direction: Conventional current is the flow of positive charge; in metals, electrons (negative) move opposite to current direction.

• Drift Velocity: Electrons move randomly but have a slow net drift opposite to $\vec{E}$ when a field is applied.

Ohm’s Law, Resistance, & Resistivity

• Ohm’s Law: $V = IR$ or $R = \frac{V}{I}$ (Units: Ohm ($\Omega$), $1\ \Omega = 1\ \mathrm{V/A}$)

• Resistivity ($\rho$): Resistance of a uniform conductor: $R = \rho \frac{L}{A}$ where $L$ is length, $A$ is cross-sectional area.

• Temperature Dependence: In metals, resistivity increases with temperature; in superconductors, resistivity drops to zero below a critical temperature $T_c$.

• Resistor Color Code: Black (0), Brown (1), Red (2), Orange (3), Yellow (4), Green (5), Blue (6), Violet (7), Gray (8), White (9). Tolerance: Gold (5%), Silver (10%), None (20%).

Electromotive Force (emf) & Source

• Electromotive Force (E): Maximum potential difference a source provides when no current flows (open circuit).

• Internal Resistance (r): Real sources have internal resistance that reduces terminal voltage when current flows.

• Terminal Voltage ($V_{ab}$):

  • Supplying current: $V_{ab} = E - Ir$

  • Charging: $V_{ab} = E + Ir$

Electrical Energy & Power

• Electrical Power: $P = V_{ab} I$ (Units: Watt (W), $1\ \mathrm{W} = 1\ \mathrm{J/s}$)

• Joule Heating: $P = I^2 R = \frac{V^2}{R}$

• Source Energy Balance: Total generated power: $P_{total} = EI$; internal dissipation: $P_{int} = I^2 r$; useful output: $P_{output} = EI - I^2 r$.

Resistor Combinations

• Series:

  • Same current through all ($I_1 = I_2 = I$).

  • Voltages add: $V_{ab} = V_1 + V_2$.

  • Equivalent resistance: $R_{eq} = R_1 + R_2 + \ldots$

• Parallel:

  • Currents add: $I = I_1 + I_2$.

  • Same voltage across all: $V_1 = V_2 = V$.

  • Equivalent resistance: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$ $R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$ (for two resistors)

Kirchhoff’s Rules

• Junction Rule (Conservation of Charge): The total current entering a junction equals the total current leaving: $\sum I_{in} = \sum I_{out}$

• Loop Rule (Conservation of Energy): The algebraic sum of potential changes around any closed loop is zero: $\sum V = 0$

• Sign Conventions:

  • Resistors: $\Delta V = -IR$ (with current), $+IR$ (against current)

  • emf Sources: $+E$ (from − to +), $-E$ (from + to −)

Measuring Instruments & RC Circuits

• Ammeter: Measures current; connected in series; low resistance.

• Voltmeter: Measures voltage; connected in parallel; high resistance.

• RC Circuit Time Constant ($\tau$): $\tau = RC$; determines charging/discharging rate.

• Capacitor Charging:

  • Charge: $q(t) = Q_f (1 - e^{-t/RC})$

  • Current: $i(t) = I_0 e^{-t/RC}$, where $I_0 = \frac{E}{R}$

  • At $t = \tau$: $q(\tau) = 0.632 Q_f$, $i(\tau) = 0.368 I_0$

Example:

In an RC circuit with $R = 1\ \mathrm{k\Omega}$ and $C = 1\ \mu\mathrm{F}$, the time constant is $\tau = RC = 1\ \mathrm{ms}$. After $1\ \mathrm{ms}$, the capacitor has charged to 63.2% of its maximum value.

Additional info: This guide covers the foundational concepts of electrostatics, electric potential, capacitance, and DC circuits, including practical aspects such as circuit analysis and measuring instruments. All equations are provided in LaTeX for clarity and further study.