Electric Charges and Fields

ELECTRIC CHARGES AND FIELDS — COMPREHENSIVE CHAPTER NOTES

1.1 INTRODUCTION TO ELECTROSTATICS

**Electrostatics** is the study of forces, fields, and potentials arising from **static charges** — charges that do not move or change with time.

Common observations of static electricity:

Spark when removing synthetic clothes in dry weather

Lightning during thunderstorms

Electric shock from door handles or bus iron bars

These occur due to accumulation and discharge of static charges through rubbing

Fundamental concept: All electrical phenomena at the macroscopic level originate from interactions between electric charges.

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1.2 ELECTRIC CHARGE

Historical Background and Discovery

The Greek philosopher **Thales of Miletus** (circa 600 BC) discovered that **amber** (elektron in Greek) rubbed with wool or silk attracts light objects like straw, pith balls, and paper bits. This phenomenon led to the term "electricity."

Experimental Observations Establishing Two Types of Charge

When various materials are rubbed:

**Two glass rods rubbed with wool/silk**: Repel each other

**Two pieces of wool or silk**: Repel each other

**Glass rod and wool**: Attract each other

**Two plastic rods rubbed with cat's fur**: Repel each other

**Plastic rod and cat's fur**: Attract each other

**Plastic rod and glass rod**: Attract each other

**Glass rod and fur**: Repel each other

Laws of Interaction Between Charges

From these observations, two fundamental laws emerged:

1. **Like charges repel each other**

2. **Unlike charges attract each other**

Neutralisation of Charge

When an electrified glass rod is brought into contact with the silk from which it was rubbed, neither body exhibits attraction or repulsion afterward. This demonstrates that **unlike charges neutralize each other**.

Benjamin Franklin's Nomenclature

American scientist **Benjamin Franklin** named the two types of charges:

**Positive charge**: Acquired by glass rod when rubbed with silk; also acquired by cat's fur when rubbed with plastic

**Negative charge**: Acquired by silk when rubbed with glass; also acquired by plastic when rubbed with cat's fur

**Convention**: By agreement, the charge on glass rubbed with silk is taken as positive; charge on plastic or silk is negative.

An **electrified/charged body** possesses an excess or deficit of charge

An **electrically neutral body** has equal positive and negative charges

Gold-Leaf Electroscope

A simple device to detect charge on a body consisting of:

A vertical metal rod housed in a box

Two thin gold leaves attached to the rod's bottom end

When a charged object touches the metal knob at the top, charge flows to the leaves

The **degree of divergence** of the leaves indicates the amount of charge

Mechanism of Charging

All matter is composed of atoms/molecules containing both positive charges (protons) and negative charges (electrons). Normally, materials are electrically neutral with balanced charges.

**During rubbing**:

Some loosely bound electrons transfer from one body to another

No new charges are created — only redistribution occurs

A body becomes **positively charged** by losing electrons

A body becomes **negatively charged** by gaining electrons

Example: When a glass rod is rubbed with silk, electrons transfer from glass to silk. Glass becomes positive; silk becomes negative.

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1.3 CONDUCTORS AND INSULATORS

Conductors

**Definition**: Materials that readily allow passage of electricity (passage of charges) through them.

Characteristic: Contain electric charges (primarily electrons) that are **comparatively free to move** inside the material.

Examples: Metals, human and animal bodies, earth, sea water

Property: **When charge is transferred to a conductor, it readily distributes over the entire surface** of the conductor.

Insulators

**Definition**: Materials that offer **high resistance** to the passage of electricity through them.

Characteristic: Charges are **tightly bound** to atoms and have minimal freedom to move.

Examples: Glass, porcelain, plastic, nylon, wood, rubber, silk

Property: **When charge is placed on an insulator, it remains at the same location** and does not redistribute.

Practical Implications

A **nylon/plastic comb** gets electrified when rubbing dry hair because plastic is an insulator and charges accumulate on the surface

A **metal spoon** does not get electrified when rubbed because it is a conductor — charges leak through our body to the ground (both are conductors)

A **metal rod with wooden/plastic handle** shows signs of charging if not touched on the metal part, because the handle (insulator) prevents charge leakage

Semiconductors

A third intermediate category exists with electrical conductivity between conductors and insulators (not emphasized for this chapter but important for semiconductor chapters).

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1.4 BASIC PROPERTIES OF ELECTRIC CHARGE

Point Charge Definition

**Point charge**: A charged body whose physical dimensions are very small compared to the distances involved in the problem.

Treatment: All charge content is assumed **concentrated at a single point in space**.

This simplification allows application of Coulomb's law and field calculations.

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1.4.1 ADDITIVITY OF CHARGES

**Principle**: Electric charges add algebraically like real numbers; they are **scalar quantities**.

**Mathematical expression**: For a system containing n charges q₁, q₂, q₃, ..., qₙ, the total charge is:

**Q_total = q₁ + q₂ + q₃ + ... + qₙ**

**Key characteristics**:

Charge has magnitude but no direction (unlike vector quantities)

Similar to mass in this respect

**Unlike mass, charge can be positive or negative** — proper algebraic signs must be used

**Example**: A system containing charges +1, +2, −3, +4, −5 (in arbitrary units) has total charge:

Q = (+1) + (+2) + (−3) + (+4) + (−5) = **−1 unit**

**Worked Example 1.1**: If 10⁹ electrons move out of a body every second, how much time is required to accumulate a total charge of 1 C on the receiving body?

**Solution**:

Charge moved per second: q = 10⁹ × e = 10⁹ × 1.6 × 10⁻¹⁹ C = 1.6 × 10⁻¹⁰ C

Time for 1 C: t = 1 C ÷ (1.6 × 10⁻¹⁰ C/s) = 6.25 × 10⁹ seconds

Converting to years: t = 6.25 × 10⁹ ÷ (365 × 24 × 3600) ≈ **198 years ≈ 200 years**

**Conclusion**: One coulomb is an extremely large charge; 10⁹ electrons per second would require ~200 years to accumulate 1 C. This demonstrates why smaller units (mC, μC) are used in practice.

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1.4.2 CONSERVATION OF CHARGE

**Statement**: In an **isolated system**, the total electric charge is always conserved. Charge can be neither created nor destroyed — only redistributed.

**Mechanism**: When two bodies are rubbed:

One body gains exactly the charge the other body loses

What one gains in positive charge, the other loses

**Within an isolated system**:

Due to interactions among bodies, charges redistribute

Total charge remains constant

**Q_initial = Q_final**

**Experimental verification**: Established through careful experiments and is now a fundamental law of nature.

**Particle Creation Exception**: In subatomic physics, charged particles can be created or destroyed in pairs with **equal and opposite charges** such that net charge remains zero.

Example: Neutron decay creates a proton and electron with charges +e and −e respectively; total charge before and after = 0.

**Worked Example 1.2**: How much positive and negative charge is contained in a cup of water of mass 250 g?

**Solution**:

Molar mass of water = 18 g/mol

Number of moles = 250 g ÷ 18 g/mol = 13.89 mol

Number of molecules = 13.89 × 6.02 × 10²³ = 8.36 × 10²⁴ molecules

Each H₂O molecule contains: 2 H atoms (2 protons, 2 electrons) + 1 O atom (8 protons, 8 electrons) = 10 protons and 10 electrons

Total electrons = 8.36 × 10²⁴ × 10 = 8.36 × 10²⁵

Total charge (negative) = 8.36 × 10²⁵ × 1.6 × 10⁻¹⁹ C = **1.34 × 10⁷ C**

Total charge (positive) = **1.34 × 10⁷ C** (equal magnitude, opposite sign)

**Conclusion**: Equal amounts of positive and negative charge exist in the cup of water, showing charge neutrality in everyday matter. The magnitudes are enormous (tens of millions of coulombs) but cancel out.

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1.4.3 QUANTISATION OF CHARGE

**Definition**: Electric charge is quantised — all free charges in the universe are **integral multiples of a basic unit of charge**.

**Mathematical expression**:

**q = ne**

where:

**q** = charge on the body

**n** = integer (positive, negative, or zero)

**e** = elementary charge = **1.602192 × 10⁻¹⁹ C** (approximately **1.6 × 10⁻¹⁹ C**)

**Charge of elementary particles**:

**Electron**: charge = −e = −1.6 × 10⁻¹⁹ C

**Proton**: charge = +e = +1.6 × 10⁻¹⁹ C

**Neutron**: charge = 0

**Meaning**: Charge cannot take arbitrary values. It must always be a whole-number multiple of e. Fractional charges do not exist (in classical physics).

Discovery and Experimental Verification

**Historical**:

First suggested by **Michael Faraday** through laws of electrolysis

Experimentally demonstrated by **Robert Millikan** in 1909-1912 using the **oil drop experiment**

Millikan measured charge on individual oil droplets and found all were integral multiples of e

SI Unit Definition

**1 Coulomb (C)** is defined as:

The charge flowing through a wire in **1 second** when the electric current is **1 ampere**

1 C = charge carried by 1 A current for 1 s

Numerical value: e = 1.6 × 10⁻¹⁹ C

**Practical units** (since 1 C is very large):

**1 millicoulomb (mC)** = 10⁻³ C

**1 microcoulomb (μC)** = 10⁻⁶ C

**1 nanocoulomb (nC)** = 10⁻⁹ C

**Example of magnitude**:

**Number of electrons in −1 C**: 1 C ÷ (1.6 × 10⁻¹⁹ C/electron) = 6.25 × 10¹⁸ electrons

Macroscopic vs. Microscopic Scale

**At macroscopic level** (charges of μC or mC):

A charge of 1 μC = 10⁻⁶ C contains approximately 10⁶ ÷ (1.6 × 10⁻¹⁹) ≈ 6.25 × 10¹² electrons

The discrete nature of charge becomes invisible at this scale

Charge appears to be **continuously distributed** (analogous to how a dotted line viewed from distance appears continuous)

**Quantisation is negligible** and charge can be treated as continuous

**At microscopic level** (charges of order ~10⁻¹⁸ C or smaller):

Individual electrons and protons are countable

The **grainy, discrete nature** of charge becomes significant

Quantisation cannot be ignored

Charge appears in discrete lumps

**Practical consequence**: For Class 12 board exam level, charge is treated as continuously distributed in electrostatics problems. Quantisation is important for understanding atomic/nuclear physics.

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1.5 COULOMB'S LAW

**Statement**: The **electrostatic force** between two point charges is:

1. Directly proportional to the **product of magnitudes** of the two charges

2. Inversely proportional to the **square of distance** between them

3. Acts along the **line joining the two charges**

4. Repulsive if charges are alike; attractive if charges are unlike

Mathematical Formulation

**Magnitude form**:

**F = k(q₁q₂)/r²**

where:

**F** = magnitude of electrostatic force (in Newtons)

**q₁, q₂** = magnitudes of point charges (in Coulombs)

**r** = distance between charges (in meters)

**k** = Coulomb's constant = 9 × 10⁹ N⋅m²⋅C⁻²

**Standard SI form** (using permittivity):

**F = (1/4πε₀) × (q₁q₂)/r²**

where:

**ε₀** = **permittivity of free space** = **8.854 × 10⁻¹² C²⋅N⁻¹⋅m⁻²**

Note: k = 1/(4πε₀)

Relationship: 4πε₀ ≈ 1.112 × 10⁻¹⁰ C²⋅N⁻¹⋅m⁻²

Historical Development: Coulomb's Experimental Method

**Apparatus**: **Torsion balance** — a sensitive device for measuring small forces

**Problem**: How to discover the law without knowing the charge magnitudes?

**Coulomb's ingenious approach**:

1. Start with a metallic sphere carrying charge q

2. Bring it in contact with an identical uncharged sphere

3. By symmetry and **additivity of charge**, each sphere now carries **q/2**

4. Separate and repeat the process to obtain **q/4, q/8**, etc.

5. For each pair, vary distance and measure force

6. Compare forces for different charge pairs at different distances

7. Conclude the relationship: F ∝ (q₁q₂)/r²

**Key insight**: The method used **charge additivity and conservation** to generate charges of known ratios without measuring absolute charge values.

Validity and Limitations

Verified experimentally from **macroscopic scale down to atomic scale** (r ~ 10⁻¹⁰ m)

Breaks down at nuclear distances (r ~ 10⁻¹⁵ m) where nuclear forces dominate

Valid for point charges or spherically symmetric charge distributions

Definition of Coulomb Unit

The choice of k = 9 × 10⁹ N⋅m²⋅C⁻² defines the SI unit coulomb:

**1 coulomb** is the charge such that two identical 1 C charges separated by 1 m experience a repulsive force of **9 × 10⁹ N**

From Coulomb's law: F = 9 × 10⁹ × (1 × 1)/(1²) = 9 × 10⁹ N

This demonstrates that **1 coulomb is an extremely large unit** — two 1 C charges 1 m apart would experience astronomical repulsive force. In practice, charges are in mC or μC range.

Vector Form of Coulomb's Law

**Notation**:

**r₁, r₂** = position vectors of charges q₁ and q₂

**r₂₁** = r₂ − r₁ = vector from charge 1 to charge 2

**r₂₁** = magnitude of r₂₁ = distance between charges

**r̂₂₁** = unit vector along r₂₁ = r₂₁/r₂₁

**Force on q₁ due to q₂**:

**F₁₂ = (1/4πε₀) × (q₁q₂/r₂₁²) × r̂₂₁**

**Force on q₂ due to q₁**:

**F₂₁ = (1/4πε₀) × (q₁q₂/r₁₂²) × r̂₁₂ = −F₁₂**

(Newton's third law: Forces are equal and opposite)

Sign Convention and Physical Interpretation

**For charges of same sign (both positive or both negative)**:

Product q₁q₂ is positive

Force is in direction of r̂₂₁ (or opposite for force on q₂)

Result: **Repulsive force**

**For charges of opposite sign**:

Product q₁q₂ is negative

Force is opposite to r̂₂₁

Result: **Attractive force**

The sign automatically handles the direction through the vector formulation.

Worked Numerical Examples

**Example 1.3**: Two point charges q₁ = +2 μC and q₂ = −3 μC are separated by 30 cm. Calculate the force between them.

**Solution**:

q₁ = +2 μC = 2 × 10⁻⁶ C

q₂ = −3 μC = −3 × 10⁻⁶ C

r = 30 cm = 0.3 m

k = 9 × 10⁹ N⋅m²⋅C⁻²

F = k|q₁||q₂|/r²

F = (9 × 10⁹) × (2 × 10⁻⁶) × (3 × 10⁻⁶) / (0.3)²

F = (9 × 10⁹) × (6 × 10⁻¹²) / 0.09

F = 54 × 10⁻³ / 0.09

F = 0.054 / 0.09

F = **0.6 N = 600 mN**

**Nature**: Attractive (opposite charges)

**Example 1.4**: Three charges are placed on a line: q₁ = +2 μC at x = 0, q₂ = +3 μC at x = 20 cm, and q₃ = −4 μC at x = 50 cm. Find the net force on q₂.

**Solution**:

Force on q₂ due to q₁:

r₁₂ = 0.2 m

F₁ = (9 × 10⁹) × (2 × 10⁻⁶) × (3 × 10⁻⁶) / (0.2)²

F₁ = (9 × 10⁹) × (6 × 10⁻¹²) / 0.04

F₁ = 54 × 10⁻³ / 0.04 = 1.35 N (repulsive, pointing right/positive direction)

Force on q₂ due to q₃:

r₂₃ = 0.5 − 0.2 = 0.3 m

F₂ = (9 × 10⁹) × (3 × 10⁻⁶) × (4 × 10⁻⁶) / (0.3)²

F₂ = (9 × 10⁹) × (12 × 10⁻¹²) / 0.09

F₂ = 108 × 10⁻³ / 0.09 = 1.2 N (attractive, pointing right/positive direction)

Net force on q₂:

F_net = F₁ + F₂ = 1.35 + 1.2 = **2.55 N (toward the right/positive direction)**

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IMPORTANT EXAM POINTS AND KEY TAKEAWAYS

1. **Charge types**: Only two types exist (positive and negative); like charges repel, unlike charges attract

2. **Charge properties**: Additive (scalar), conserved, quantised (q = ne)

3. **Conductors vs insulators**: Charges redistribute in conductors, remain localized in insulators

4. **Coulomb's law**: F ∝ q₁q₂/r², acts along line of centers, vector form handles direction automatically

5. **Permittivity**: ε₀ = 8.854 × 10⁻¹² C²⋅N⁻¹⋅m⁻²; k = 1/(4πε₀) = 9 × 10⁹ N⋅m²⋅C⁻²

6. **Coulomb unit definition**: 1 C defined such that two 1 C charges 1 m apart experience 9 × 10⁹ N force

7. **Macroscopic charge**: Treated as continuous despite quantisation

8. **Superposition principle**: Total force on a charge = vector sum of forces from all other charges