Formula Sheet: Electrostatics

Electric Charge

Fundamental Properties

• Elementary charge: \( e = 1.6 \times 10^{-19} \, \text{C} \)
• Charge quantization: \( Q = ne \), where \( n \) is an integer
• Conservation of charge: Total charge in an isolated system remains constant
• Types of charge: Positive (+) and negative (-)

Coulomb's Law

Force Between Point Charges

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

• \( F \) = electrostatic force (N)
• \( k \) = Coulomb's constant = \( 9.0 \times 10^9 \, \text{N·m}^2/\text{C}^2 \)
• \( q_1, q_2 \) = magnitudes of the two charges (C)
• \( r \) = distance between the charges (m)
• Direction: Along the line connecting the charges (attractive for opposite charges, repulsive for like charges)

Coulomb's Constant and Permittivity

\[ k = \frac{1}{4\pi\epsilon_0} \]

• \( \epsilon_0 \) = permittivity of free space = \( 8.85 \times 10^{-12} \, \text{C}^2/(\text{N·m}^2) \)

Vector Form of Coulomb's Law

\[ \vec{F} = k \frac{q_1 q_2}{r^2} \hat{r} \]

• \( \hat{r} \) = unit vector pointing from one charge to the other
• Note: Sign of charges determines direction (positive force = repulsion, negative = attraction)

Superposition Principle

\[ \vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \cdots \]

• The net force on a charge is the vector sum of all individual forces
• Forces from multiple charges add vectorially

Electric Field

Definition and Fundamental Relationships

\[ \vec{E} = \frac{\vec{F}}{q_0} \]

• \( \vec{E} \) = electric field (N/C or V/m)
• \( \vec{F} \) = force on test charge (N)
• \( q_0 \) = small positive test charge (C)
• Direction: Direction a positive test charge would experience force

Electric Field of a Point Charge

\[ E = k \frac{|q|}{r^2} \]

• \( E \) = magnitude of electric field (N/C)
• \( q \) = source charge creating the field (C)
• \( r \) = distance from the source charge (m)
• Direction: Away from positive charges, toward negative charges

Force on a Charge in an Electric Field

\[ \vec{F} = q\vec{E} \]

• \( q \) = charge experiencing the force (C)
• \( \vec{E} \) = electric field at the charge's location (N/C)
• Note: Positive charges experience force in direction of field; negative charges opposite

Electric Field Superposition

\[ \vec{E}_{\text{net}} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \cdots \]

• Electric fields from multiple sources add vectorially

Electric Field Between Parallel Plates

\[ E = \frac{\sigma}{\epsilon_0} \]

• \( \sigma \) = surface charge density (C/m²)
• Note: Field is uniform between plates, pointing from positive to negative plate
• Simplified form: For capacitors, often written as \( E = \frac{V}{d} \)

Electric Potential Energy

Potential Energy Between Two Point Charges

\[ U = k \frac{q_1 q_2}{r} \]

• \( U \) = electric potential energy (J)
• \( q_1, q_2 \) = charges (C) - signs matter!
• \( r \) = separation distance (m)
• Convention: \( U = 0 \) at \( r = \infty \)
• Sign convention: Positive U for like charges (repulsive), negative U for opposite charges (attractive)

Change in Potential Energy

\[ \Delta U = -W_{\text{field}} \]

• \( W_{\text{field}} \) = work done by the electric field (J)
• Positive work by field decreases potential energy

Work-Energy Relationship

\[ W_{\text{external}} = \Delta U \]

• \( W_{\text{external}} \) = work done by external force against the field (J)

Potential Energy in a Uniform Field

\[ U = qEd \]

• \( q \) = charge (C)
• \( E \) = uniform electric field (N/C)
• \( d \) = displacement in direction of field (m)
• Note: Assumes reference point at initial position

Electric Potential (Voltage)

Definition

\[ V = \frac{U}{q} = \frac{W}{q} \]

• \( V \) = electric potential (V or J/C)
• \( U \) = potential energy (J)
• \( q \) = test charge (C)
• Note: Scalar quantity (no direction)

Potential of a Point Charge

\[ V = k \frac{q}{r} \]

• \( V \) = electric potential (V)
• \( q \) = source charge (C) - sign matters!
• \( r \) = distance from charge (m)
• Convention: \( V = 0 \) at \( r = \infty \)

Potential Difference (Voltage)

\[ \Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{s} \]

• \( \Delta V \) = potential difference (V)
• Simplified for uniform field: \( \Delta V = Ed \)

Potential Difference in Uniform Field

\[ \Delta V = Ed \]

• \( E \) = uniform electric field strength (N/C or V/m)
• \( d \) = distance moved in direction of field (m)
• Note: Potential decreases in direction of electric field

Relationship Between Field and Potential

\[ E = -\frac{dV}{dr} \]

• Electric field points in direction of decreasing potential
• For uniform field: \( E = -\frac{\Delta V}{\Delta r} \)

Superposition of Potentials

\[ V_{\text{net}} = V_1 + V_2 + V_3 + \cdots \]

• Potentials are scalars and add algebraically (not vectorially)

Work and Energy in Electric Fields

Work Done by Electric Field

\[ W_{\text{field}} = q \Delta V = q(V_A - V_B) \]

• \( q \) = charge being moved (C)
• \( V_A \) = potential at starting point (V)
• \( V_B \) = potential at ending point (V)
• Note: Work is positive when charge moves to lower potential for positive charge

Work Done Against Electric Field

\[ W_{\text{external}} = -W_{\text{field}} = q(V_B - V_A) = q\Delta V \]

Change in Kinetic Energy

\[ \Delta KE = W_{\text{field}} = q(V_A - V_B) \]

• Conservation of energy: \( \Delta KE + \Delta U = 0 \) (no external work)

Electron-Volt (eV)

\[ 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \]

• Energy gained by electron moving through potential difference of 1 V
• \( KE = q\Delta V \) where \( q = e \)

Capacitance

Definition of Capacitance

\[ C = \frac{Q}{V} \]

• \( C \) = capacitance (F, farads)
• \( Q \) = magnitude of charge on each plate (C)
• \( V \) = potential difference between plates (V)
• Note: 1 F = 1 C/V

Parallel Plate Capacitor

\[ C = \frac{\epsilon_0 A}{d} \]

• \( A \) = area of each plate (m²)
• \( d \) = separation between plates (m)
• \( \epsilon_0 \) = permittivity of free space
• Note: Valid for \( d \ll \) plate dimensions

Capacitance with Dielectric

\[ C = \kappa C_0 = \frac{\kappa \epsilon_0 A}{d} \]

• \( \kappa \) = dielectric constant (dimensionless, \( \kappa \geq 1 \))
• \( C_0 \) = capacitance without dielectric (F)
• Note: Dielectric increases capacitance

Energy Stored in a Capacitor

\[ U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C} \]

• \( U \) = energy stored (J)
• All three forms are equivalent - use whichever is most convenient

Energy Density in Electric Field

\[ u = \frac{1}{2}\epsilon_0 E^2 \]

• \( u \) = energy per unit volume (J/m³)
• \( E \) = electric field strength (V/m)
• With dielectric: \( u = \frac{1}{2}\kappa\epsilon_0 E^2 \)

Capacitors in Circuits

Capacitors in Parallel

\[ C_{\text{eq}} = C_1 + C_2 + C_3 + \cdots \]

• Equivalent capacitance is sum of individual capacitances
• Same voltage across all capacitors
• Charges add: \( Q_{\text{total}} = Q_1 + Q_2 + Q_3 + \cdots \)

Capacitors in Series

\[ \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \cdots \]

• Reciprocals add
• Same charge on all capacitors
• Voltages add: \( V_{\text{total}} = V_1 + V_2 + V_3 + \cdots \)

Two Capacitors in Series (Simplified)

\[ C_{\text{eq}} = \frac{C_1 C_2}{C_1 + C_2} \]

Electric Dipole

Dipole Moment

\[ p = qd \]

• \( p \) = dipole moment (C·m)
• \( q \) = magnitude of each charge (C)
• \( d \) = separation between charges (m)
• Direction: From negative to positive charge

Torque on Dipole in Electric Field

\[ \tau = pE\sin\theta \]

• \( \tau \) = torque (N·m)
• \( \theta \) = angle between dipole moment and electric field
• Vector form: \( \vec{\tau} = \vec{p} \times \vec{E} \)

Potential Energy of Dipole in Electric Field

\[ U = -pE\cos\theta = -\vec{p} \cdot \vec{E} \]

• Minimum energy: When dipole aligned with field (\( \theta = 0° \))
• Maximum energy: When dipole opposes field (\( \theta = 180° \))

Gauss's Law

General Form

\[ \Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \]

• \( \Phi_E \) = electric flux (N·m²/C or V·m)
• \( Q_{\text{enc}} \) = charge enclosed by Gaussian surface (C)
• \( \oint \) = closed surface integral
• \( d\vec{A} \) = differential area element (perpendicular to surface)

Electric Flux

\[ \Phi_E = \vec{E} \cdot \vec{A} = EA\cos\theta \]

• \( \theta \) = angle between field and area vector
• Uniform field, flat surface only

Applications of Gauss's Law

Spherical Symmetry (Point Charge or Sphere)

\[ E = k\frac{Q}{r^2} \]

• Outside charged sphere: same as point charge at center
• Inside uniformly charged sphere: \( E = 0 \) (for conducting sphere)

Infinite Line of Charge

\[ E = \frac{\lambda}{2\pi\epsilon_0 r} = \frac{2k\lambda}{r} \]

• \( \lambda \) = linear charge density (C/m)
• \( r \) = perpendicular distance from line (m)

Infinite Plane of Charge

\[ E = \frac{\sigma}{2\epsilon_0} \]

• \( \sigma \) = surface charge density (C/m²)
• Note: Field independent of distance from plane

Conductors in Electrostatic Equilibrium

Properties of Conductors

• Electric field inside conductor: \( E = 0 \)
• Net charge inside conductor: \( Q_{\text{interior}} = 0 \)
• Excess charge: Resides on outer surface
• Electric field at surface: Perpendicular to surface
• Potential throughout conductor: Constant (equipotential)

Electric Field at Conductor Surface

\[ E = \frac{\sigma}{\epsilon_0} \]

• \( \sigma \) = local surface charge density (C/m²)
• Direction: Perpendicular to surface

Important Constants and Conversions

• Coulomb's constant: \( k = 9.0 \times 10^9 \, \text{N·m}^2/\text{C}^2 \)
• Permittivity of free space: \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/(\text{N·m}^2) \)
• Elementary charge: \( e = 1.6 \times 10^{-19} \, \text{C} \)
• Electron-volt: \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \)
• Relation: \( k = \frac{1}{4\pi\epsilon_0} \)

Key Relationships and Conversions

• Electric field units: 1 N/C = 1 V/m
• Potential energy and voltage: \( U = qV \)
• Field and potential: \( E = -\frac{dV}{dr} \) or \( \Delta V = -Ed \) (uniform field)
• Force and field: \( F = qE \)
• Capacitance units: 1 F = 1 C/V = 1 C²/J
• Common prefixes: 1 μF = 10^-6 F, 1 pF = 10^-12 F, 1 nF = 10^-9 F