CBSE Class 12 Physics Electric Charges and Fields Notes Set 03

1. Electric Charge

Electric charge is a basic property of matter that makes it experience a force in an electric field. There are two kinds of charge, called positive and negative. Like charges repel and unlike charges attract. The SI unit of charge is the coulomb (C).

(i) Charging by friction: When two suitable bodies are rubbed together, electrons move from one body to the other. A glass rod rubbed with silk loses electrons and becomes positive; the silk gains them and becomes equally negative.
(ii) Conductors and insulators: Conductors (metals, the human body, the earth) contain free electrons and let charge move through them. Insulators (glass, plastic, dry wood) hold charge where it is placed.

2. Basic Properties of Electric Charge

(i) Additivity: Charges add like real numbers (scalars). A system with charges q₁, q₂, …, q_n has total charge q = q₁ + q₂ + … + q_n, taking signs into account.
(ii) Conservation: The total charge of an isolated system never changes. Charge can be transferred from one body to another, but it cannot be created or destroyed.
(iii) Quantisation: The charge on any body is an integral multiple of the basic charge e.
q = ne, where n = 0, ±1, ±2, ±3, … and e = 1.6 × 10⁻¹⁹ C
At the macroscopic level the charges are so large compared with e that quantisation can be ignored and charge treated as continuous.

3. Coulomb's Law

The force between two stationary point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them, and acts along the line joining them.
F = \( \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2} \) = k \( \frac{q_1 q_2}{r^2} \)
k = \( \frac{1}{4\pi\varepsilon_0} \) = 9 × 10⁹ N m² C⁻² and ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻² (permittivity of free space)

Vector form: The force on q₂ due to q₁ is
\( \vec{F}_{21} = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r_{21}^2} \hat{r}_{21} \), where \( \hat{r}_{21} \) is the unit vector from q₁ to q₂.
\( \vec{F}_{12} = -\vec{F}_{21} \), so Coulomb forces obey Newton's third law.

Force in a medium: In a medium of dielectric constant (relative permittivity) K,
F_medium = \( \frac{F_{\text{vacuum}}}{K} \), where K = \( \frac{\varepsilon}{\varepsilon_0} \). For water K ≈ 80, so the force between two charges in water is about 80 times weaker than in air.

Comparison with gravitation: Both forces follow an inverse square law and act along the line joining the bodies. The Coulomb force can be attractive or repulsive and is far stronger: between two protons it is about 10³⁶ times the gravitational force.

4. Forces between Multiple Charges (Superposition Principle)

The force on any charge due to a number of other charges is the vector sum of the forces due to each charge taken separately. Each pairwise force is unaffected by the presence of the other charges.
\( \vec{F}_1 = \vec{F}_{12} + \vec{F}_{13} + \ldots + \vec{F}_{1n} \)

5. Electric Field

The electric field at a point is the force per unit positive test charge placed at that point. The test charge is taken very small so that it does not disturb the charges producing the field.
\( \vec{E} = \lim_{q_0 \to 0} \frac{\vec{F}}{q_0} \)
Field of a point charge q at distance r: E = \( \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \), directed away from a positive charge and towards a negative charge.
SI unit: N C⁻¹ (or V m⁻¹). A charge q in a field E experiences a force F = qE.
The field of a group of charges is the vector sum of the fields of the individual charges (superposition).

6. Electric Field Lines

An electric field line is a curve drawn so that the tangent at each point gives the direction of the field at that point.

Properties of electric field lines:
1. They start from positive charges and end on negative charges (or at infinity for a single charge).
2. Two field lines never cross, because the field cannot have two directions at one point.
3. Electrostatic field lines do not form closed loops.
4. Where the lines are crowded the field is strong; where they are spread out the field is weak.
5. In a uniform field the lines are parallel and equally spaced.
6. Field lines meet the surface of a conductor at right angles.

7. Electric Flux

Electric flux through a surface measures the number of field lines passing through it.
For a small area ΔS in a field E: Δφ = \( \vec{E} \cdot \Delta\vec{S} \) = E ΔS cos θ, where θ is the angle between E and the normal to the area.
Total flux: φ = \( \oint \vec{E} \cdot d\vec{S} \)
Flux is a scalar. SI unit: N m² C⁻¹ (or V m). It is maximum when the area is perpendicular to the field (θ = 0°) and zero when the area is parallel to it (θ = 90°).

8. Electric Dipole

A pair of equal and opposite charges +q and −q separated by a small distance 2a forms an electric dipole.
Dipole moment: p = q × 2a, a vector directed from −q to +q along the dipole axis. SI unit: C m.

(i) Field on the axial line at distance r from the centre:
E = \( \frac{1}{4\pi\varepsilon_0} \frac{2pr}{(r^2 - a^2)^2} \), which for r ≫ a becomes E = \( \frac{1}{4\pi\varepsilon_0} \frac{2p}{r^3} \), in the direction of p.
(ii) Field on the equatorial line at distance r from the centre:
E = \( \frac{1}{4\pi\varepsilon_0} \frac{p}{(r^2 + a^2)^{3/2}} \), which for r ≫ a becomes E = \( \frac{1}{4\pi\varepsilon_0} \frac{p}{r^3} \), opposite to p.
(iii) At the same large distance, E_axial = 2 E_equatorial. The field of a dipole falls as 1/r³, faster than the 1/r² field of a single charge.

9. Dipole in a Uniform External Field

The forces qE and −qE on the two charges are equal and opposite, so the net force is zero. They act at different points, so they form a couple.
Torque: \( \vec{\tau} = \vec{p} \times \vec{E} \), magnitude τ = pE sin θ
1. τ is maximum (= pE) when θ = 90°.
2. τ = 0 when θ = 0° (stable equilibrium) or θ = 180° (unstable equilibrium).
3. The torque tends to turn the dipole into line with the field.
In a non-uniform field the forces on the two charges are unequal, so the dipole feels a net force as well as a torque. This is why a charged comb attracts small uncharged bits of paper: it induces dipoles in them and its field is non-uniform.

10. Continuous Charge Distributions

(i) Linear charge density: λ = \( \frac{\Delta q}{\Delta l} \), unit C m⁻¹ (charge on a wire or thin rod)
(ii) Surface charge density: σ = \( \frac{\Delta q}{\Delta S} \), unit C m⁻² (charge on a sheet or the surface of a conductor)
(iii) Volume charge density: ρ = \( \frac{\Delta q}{\Delta V} \), unit C m⁻³ (charge spread through a volume)

11. Gauss's Law

The total electric flux through any closed surface is \( \frac{1}{\varepsilon_0} \) times the total charge enclosed by it.
\( \oint \vec{E} \cdot d\vec{S} = \frac{q_{\text{enclosed}}}{\varepsilon_0} \)
1. The closed surface used is called a Gaussian surface; it can have any shape, but it should not pass through a discrete charge.
2. Charges outside the surface add nothing to the net flux, though they do affect the field at points on the surface.
3. If the net charge enclosed is zero (for example, a whole dipole), the net flux is zero.
4. Gauss's law is true for every closed surface, but it is useful for finding E only when the charge distribution is symmetric.

12. Applications of Gauss's Law

(i) Infinitely long straight uniformly charged wire (linear charge density λ), using a cylindrical Gaussian surface of radius r:
E × 2πrl = \( \frac{\lambda l}{\varepsilon_0} \)  ⇒  E = \( \frac{\lambda}{2\pi\varepsilon_0 r} \), directed radially. So E ∝ 1/r.

(ii) Uniformly charged infinite plane sheet (surface charge density σ), using a cylinder that crosses the sheet:
2EA = \( \frac{\sigma A}{\varepsilon_0} \)  ⇒  E = \( \frac{\sigma}{2\varepsilon_0} \), perpendicular to the sheet and independent of the distance from it.

(iii) Uniformly charged thin spherical shell (radius R, total charge q), using a concentric spherical Gaussian surface of radius r:
1. Outside (r > R): E = \( \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \), as if the whole charge were at the centre.
2. On the surface (r = R): E = \( \frac{1}{4\pi\varepsilon_0} \frac{q}{R^2} \) = \( \frac{\sigma}{\varepsilon_0} \)
3. Inside (r < R): E = 0, because the Gaussian surface encloses no charge.

13. Quick Formula Recap