Chapter 1 - Electric Charges and Fields

Electric charge is the intrinsic property associated with matter due to which it produces and experiences electrical and magnetic effects. There are two types of charges that exist in nature. Benjamin Franklin named them positive and negative. Convention: When a glass rod is rubbed with silk, glass gets positive charge and silk gets negative. When ebonite (hard rubber) rod is rubbed with fur, ebonite gets negative charge and fur gets positive. This happens because rubbing transfers electrons from one material to another.

Like charges repel each other and unlike charges attract each other. This is the most basic rule. If you bring two glass rods rubbed with silk near each other, they repel. Glass rod and ebonite rod attract. You can test with a pith ball electroscope - a light pith ball suspended by a thread. A charged rod attracts a neutral pith ball first by induction, then after contact the pith ball gets same charge and repels.

SI unit of charge is Coulomb (C). One Coulomb is a large amount of charge. The smallest charge is that of an electron or proton, e = 1.6 × 10⁻¹⁹ C. Charge is a scalar but has positive and negative sign.

Based on ability to allow charge or electric current to flow, materials are classified as conductors and insulators. Conductors have a large number of free electrons. Metals like copper, aluminum, silver are good conductors. When you give charge to a conductor, it spreads over the entire surface quickly.

Insulators have almost no free electrons. Glass, rubber, wood, plastic are insulators. Charge given to an insulator stays at the place where it was given, it does not spread. There is an intermediate category called semiconductors (silicon, germanium) whose conductivity lies between conductors and insulators and increases with temperature.

When we say a body is charged, we mean it has extra electrons or deficit of electrons. A neutral body has equal number of protons and electrons. If electrons are removed, body becomes positively charged. If electrons are added, body becomes negatively charged. Protons never move in charging process, only electrons move because they are much lighter and free.

There are two ways to charge a body: conduction (by contact) and induction (without contact). In conduction, you touch a charged body to a neutral body and some charge transfers. Both end up with same type of charge.

In induction, you do not touch. Bring a positively charged rod near a neutral metal sphere. The free electrons in the sphere are attracted towards the rod, so the side near the rod becomes negative and far side becomes positive. Now connect the sphere to ground (earth) with a wire. Electrons flow from ground to the positive side. Remove the ground connection while the charged rod is still nearby. Now remove the rod. The sphere is left with net negative charge. Notice we charged the sphere with opposite charge without touching the rod.

Gold leaf electroscope is used to detect charge. It has a metal rod with two thin gold leaves at bottom inside a glass case. When a charged body touches the top knob, charge spreads to leaves and leaves diverge due to repulsion. More divergence means more charge.

1. Additivity: Total charge of a system is the algebraic sum of individual charges. Charges are scalars, so you add with sign. If system has charges +2C, -3C, +4C, total = +3C. This means charge is additive.

2. Conservation: In an isolated system, total charge is conserved. Charge can neither be created nor destroyed, it can only be transferred from one body to another. When you rub glass and silk, glass becomes +q and silk becomes -q, total remains zero. In nuclear reactions, pair production and annihilation also conserve charge.

3. Quantisation: Charge is quantized. It exists in discrete packets. Total charge on any body is integral multiple of elementary charge e. q = ±ne where n is integer 0,1,2,... and e=1.6×10⁻¹⁹ C is magnitude of charge on electron. You cannot have 0.5e or 1.5e charge. This was proved by Millikan oil drop experiment. At macroscopic level, charge appears continuous because e is very small and large number of electrons are involved.

Other property: Charge is invariant - it does not depend on speed of charge. Mass depends on speed, charge does not.

Coulomb's law gives force between two point charges at rest. Statement: Force between two point charges is directly proportional to product of magnitudes of charges and inversely proportional to square of distance between them, and acts along line joining charges.

Mathematically: F = k |q1 q2| / r² where k = 1/(4πϵ0) = 9×10⁹ Nm²/C². ϵ0 = 8.85×10⁻¹² C²/Nm² is permittivity of free space. Vector form: F12 = (1/4πϵ0) q1 q2 r12_hat / r². F12 is force on q1 due to q2, r12_hat is unit vector from q2 to q1. If q1 q2 >0 (like charges) force is repulsive, if <0 attractive.

Force obeys Newton's third law: F12 = -F21 equal and opposite. Force depends on medium. In medium other than vacuum, force becomes Fm = F_vacuum / εr where εr = ε/ε0 is relative permittivity or dielectric constant of medium. For water εr≈80, so force becomes 80 times weaker in water.

Coulomb's law is valid for point charges, distance much larger than size of charges, and charges at rest (electrostatics). For larger distances and moving charges, it is still approximately valid.

Example: Force between two small charged spheres charges 2×10⁻⁷ C and 3×10⁻⁷ C placed 30 cm apart in air: F = 9e9 *2e-7*3e-7 /0.3² = 6×10⁻³ N repulsive.

What if more than two charges? Superposition principle says force on any charge due to a number of other charges is the vector sum of forces due to each charge taken individually. So if you have charges q1, q2, q3... and want force on q1, calculate F12 (due to q2), F13 (due to q3), F14 etc and add as vectors: F1 = F12 + F13 + F14 + ...

Each pair obeys Coulomb's law. Superposition is important because electric force is a two-body interaction. The presence of third charge does not affect force between first two. This allows us to calculate net force in any arrangement.

Consider three charges at corners of equilateral triangle. Force on one corner is vector sum of forces due to other two corners. You need to resolve into x and y components and add.

Continuous charge distribution: If charge is not point but spread over line, surface or volume, we define linear charge density λ = dq/dl (C/m), surface charge density σ = dq/dA (C/m²), volume charge density ρ = dq/dV (C/m³). Then small charge dq produces field and we integrate over entire distribution.

How does a charge feel force from another charge without touching? The idea of electric field solves this. A charge q creates an electric field in space around it. Another charge q0 placed in this field experiences force F = q0 E. So field is intermediary.

Definition: Electric field at a point is force per unit positive test charge placed at that point: E = F / q0. Test charge q0 must be small so it does not disturb the source charge distribution. SI unit N/C or V/m. It is a vector quantity.

Electric field due to point charge q at distance r: E = (1/4πϵ0) q / r², direction away from positive charge and towards negative charge. Magnitude falls as 1/r² similar to Coulomb force. Field exists even without test charge, it is property of space due to source charge.

Physical significance: Field is real, it has energy, it can propagate as electromagnetic wave. Faraday introduced field concept. Superposition applies to fields also: net field due to multiple charges is vector sum of fields due to each charge individually: E = E1+E2+E3+...

Example: Field due to point charge 2μC at distance 0.5m: E=9e9*2e-6/0.25=72000 N/C.

Michael Faraday introduced electric field lines as imaginary lines to visualize electric field. Properties:

1. Field line is a curve whose tangent at any point gives direction of electric field at that point.
2. Field lines start from positive charge and end on negative charge. For isolated positive, they go to infinity. For isolated negative, they come from infinity.
3. Density of field lines (number per unit area perpendicular) represents strength of field. Crowded lines strong field, sparse weak field.
4. Two field lines never intersect. If they intersect, at intersection point there would be two directions of field which is impossible.
5. Field lines do not form closed loops. Electrostatic field lines start and end on charges, they are not closed. (Magnetic field lines do form closed loops).
6. Field lines are continuous and exist in three dimensions.
7. In region with no charge, field lines are continuous. They break only at charges.

Patterns: Single positive radial outward, single negative radial inward, dipole lines start from positive and end on negative, two positive charges repel lines bend away, uniform field parallel equally spaced lines.

Electric flux measures number of electric field lines crossing a surface. It is defined as φ_E = E·A = EA cosθ where A is area vector. Area vector magnitude = area, direction = outward normal perpendicular to surface.

If E is parallel to A (θ=0°), flux maximum φ=EA. If E perpendicular to A (θ=90°), flux zero because no lines crossing. If E opposite to A (θ=180°), flux negative φ=-EA. SI unit Nm²/C or V·m.

For non-uniform field or curved surface, flux φ = ∫ E·dA integration over whole surface, dA small area element with outward normal.

Consider a cube placed in uniform field. Flux entering left face negative because θ=180°, flux leaving right face positive θ=0°, net flux through cube zero if no charge inside because entering = leaving. Flux only non-zero if charge enclosed.

Flux is scalar but can be positive (outgoing) or negative (incoming).

An electric dipole is a pair of equal and opposite charges +q and -q separated by small distance 2a. The line joining charges is dipole axis. Dipole moment is defined as p = q × 2a, vector from -q to +q. Magnitude p = q*2a, direction from negative to positive. SI unit C·m. It is a vector.

Field on axial line: Point on dipole axis (along line joining charges). E_axial = (1/4πϵ0) * 2pr / (r²-a²)². For far distance r>>a, E_axial ≈ (1/4πϵ0) * 2p / r³ along p direction to the right if +q on right.

Field on equatorial line: Point on perpendicular bisector of dipole. E_equatorial = (1/4πϵ0) * p / (r²+a²)^(3/2). For r>>a, E_equatorial ≈ (1/4πϵ0) * p / r³ opposite to p direction to left. Magnitude axial is twice equatorial: E_axial ≈ 2 × E_equatorial.

Dipole in uniform electric field: If dipole placed in uniform external field E at angle θ between p and E, it experiences torque τ = p × E magnitude τ = pE sinθ. This torque tends to rotate dipole to align p with E. Clockwise if p above E. Force on dipole in uniform field net zero (equal opposite forces) but torque non-zero. In non-uniform field, net force also exists, dipole moves towards stronger field.

Potential energy of dipole in uniform field U = -p·E = -pE cosθ. Minimum when θ=0° p aligned with E stable equilibrium, maximum when θ=180° opposite unstable.

Examples of dipoles: Water molecule H2O has dipole moment, HCl, etc.

We have seen point charges and discrete charges. When charge is spread continuously, we use charge densities: linear λ = dq/dl C/m for wire, surface σ = dq/dA C/m² for sheet, volume ρ = dq/dV C/m³ for volume. Total charge Q=∫λ dl or ∫σ dA or ∫ρ dV.

Gauss Law: One of most powerful laws in electrostatics. Statement: Total electric flux through a closed surface (Gaussian surface) is equal to net charge enclosed divided by ϵ0. φ = ∮ E·dA = Q_enclosed / ϵ0. Flux depends only on enclosed charge, not on shape or size or charges outside surface.

Gaussian surface is imaginary closed surface chosen with symmetry to make calculation easy: spherical for point charge/sphere, cylindrical for line charge, pillbox (short cylinder) for plane sheet.

Applications:

1. Infinite line charge: λ linear charge density C/m, symmetry cylindrical Gaussian surface radius r length l. Field radial outward, magnitude E = λ / (2πϵ0 r). Falls as 1/r not 1/r². Direction away from line if λ positive.

2. Infinite plane sheet: σ surface charge density C/m², Gaussian pillbox crossing sheet. Field uniform both sides perpendicular to sheet, magnitude E = σ / (2ϵ0) independent of distance. Direction away from sheet if σ positive both sides.

3. Two parallel sheets: One +σ one -σ (like capacitor) field between sheets E = σ/ϵ0 from + to -, outside zero because fields cancel.

4. Spherical shell: Thin spherical shell radius R charge Q uniformly distributed. Outside (r>R) field as point charge E = (1/4πϵ0) Q / r². Inside (r

5. Solid sphere: Uniform volume charge, outside same as point, inside E ∝ r.

Gauss law is useful only when high symmetry exists. For irregular shapes, Coulomb law superposition needed.