Electrostatics

Misconceptions

Key Concepts

Electric Charge

The fundamental property of matter that causes electromagnetic interactions.

Properties:

• •Two types: positive (+) and negative (-)
• •Like charges repel, opposite charges attract
• •Charge is quantized: q = ne, where e = 1.602 × 10⁻¹⁹ C
• •Charge is conserved in all processes
• •Unit: Coulomb (C)

Coulomb's Law

The electric force between two point charges:

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

Where:

• •k = 8.99 × 10⁹ N·m²/C² (Coulomb's constant)
• •q₁, q₂ = charges (in Coulombs)
• •r = distance between charges (in meters)
• •F = force (in Newtons)

Direction:

• •Positive F: repulsive (like charges)
• •Negative F: attractive (opposite charges)
• •Force acts along the line connecting the charges

Electric Field

A vector field that describes the electric force per unit positive test charge:

$$E = \frac{F}{q} = k\frac{Q}{r^2}$$

Where:

• •E = electric field strength (N/C or V/m)
• •Q = source charge creating the field
• •r = distance from the source charge

Direction:

• •Points away from positive charges
• •Points toward negative charges
• •At any point, E indicates the direction a positive test charge would accelerate

Electric Potential

The electric potential energy per unit charge (scalar quantity):

$$V = k\frac{Q}{r}$$

Where:

• •V = electric potential (Volts, V = J/C)
• •Q = source charge
• •r = distance from the source charge

Key points:

• •Positive charges create positive potential
• •Negative charges create negative potential
• •Potential is a scalar (no direction)
• •Potential difference (voltage) drives current

Electric Potential Energy

The energy stored in a system of charges:

$$PE = k\frac{q_1 q_2}{r} = qV$$

Where:

• •PE = potential energy (Joules)
• •q₁, q₂ = the two charges
• •r = separation distance
• •V = potential at the location of charge q

Sign convention:

• •Positive PE: like charges (repulsive, energy required to bring together)
• •Negative PE: opposite charges (attractive, energy released when brought together)

Work Done by Electric Field

Work done by the electric field when moving a charge:

$$W = q(V_i - V_f) = -\Delta PE$$

Where:

• •W = work done by the field (Joules)
• •q = charge being moved
• •Vᵢ, Vf = initial and final potentials

Key points:

• •Field does positive work when moving + charge to lower potential
• •Field does positive work when moving - charge to higher potential
• •Work by field = negative of change in PE

Conductors vs Insulators

Conductors:

• •Allow charges to move freely
• •Excess charge resides on the surface
• •Electric field inside is zero (electrostatic equilibrium)
• •Examples: metals, ionic solutions

Insulators:

• •Charges cannot move freely
• •Charge remains where placed
• •Can be polarized (induced dipoles)
• •Examples: rubber, glass, plastic

Equations

[1] F = kq₁q₂/r² (Coulomb's law)
[2] E = F/q = kQ/r² (electric field from point charge)
[3] V = kQ/r (electric potential from point charge)
[4] PE = kq₁q₂/r = qV (electric potential energy)
[5] W = q(Vᵢ - Vf) = -ΔPE (work by electric field)
[6] k = 8.99 × 10⁹ N·m²/C² (Coulomb's constant)
[7] e = 1.602 × 10⁻¹⁹ C (elementary charge)

Worked Examples

Example 1: Coulomb's Law

Problem: Two charges, q₁ = +3 μC and q₂ = -5 μC, are separated by 0.2 m. What is the magnitude of the electric force between them?

Solution:

1. •Convert units: q₁ = 3 × 10⁻⁶ C, q₂ = 5 × 10⁻⁶ C
2. •Apply Coulomb's law: F = k|q₁q₂|/r²
3. •F = (8.99 × 10⁹)(3 × 10⁻⁶)(5 × 10⁻⁶)/(0.2)²
4. •F = (8.99 × 10⁹)(15 × 10⁻¹²)/(0.04)
5. •F = 3.37 N (attractive)

Example 2: Electric Field

Problem: What is the electric field 0.5 m from a +8 μC point charge?

Solution:

1. •E = kQ/r²
2. •E = (8.99 × 10⁹)(8 × 10⁻⁶)/(0.5)²
3. •E = (8.99 × 10⁹)(8 × 10⁻⁶)/0.25
4. •E = 2.88 × 10⁵ N/C (pointing away from the charge)

Example 3: Electric Potential Energy

Problem: How much work is required to bring a +2 μC charge from infinity to a point 0.3 m from a +6 μC charge?

Solution:

1. •Work required = ΔPE = PEf - PEi
2. •PEi = 0 (at infinity)
3. •PEf = kq₁q₂/r = (8.99 × 10⁹)(2 × 10⁻⁶)(6 × 10⁻⁶)/0.3
4. •PEf = (8.99 × 10⁹)(12 × 10⁻¹²)/0.3
5. •Work = 0.36 J

Explanation Patterns

1. •Identify the charges - What are their signs and magnitudes?
2. •Determine what's asked - Force, field, potential, or energy?
3. •Draw a diagram - Show charges, distances, and direction of fields/forces
4. •Choose the appropriate equation:
   • •Force between charges: Coulomb's law
   • •Field from a charge: E = kQ/r²
   • •Potential from a charge: V = kQ/r
   • •Energy of configuration: PE = kq₁q₂/r
5. •Watch units - Convert μC to C, cm to m
6. •Check signs - Like charges repel, opposite attract

Common Problem Types

1. •Coulomb force calculations: Find force magnitude and direction between point charges
2. •Electric field from point charges: Calculate field strength and direction
3. •Superposition of fields: Add vector fields from multiple charges
4. •Electric potential calculations: Find potential at a point from one or more charges
5. •Potential energy of charge systems: Calculate energy stored in charge configurations
6. •Work-energy problems: Find work to move charges between points
7. •Conductor/insulator concepts: Qualitative questions about charge distribution