JEE Physics Electric Charges and Fields Notes Set 05

Electric Charge and Its Properties

Charge is the property of matter that gives rise to electric and magnetic effects. Its SI unit is the coulomb; 1 C = 3 × 10⁹ esu. Charge is a scalar that adds algebraically, comes in two kinds (positive means a deficiency of electrons, negative an excess), and its dimensions are [AT].
Charge is conserved in an isolated system, quantised as \( Q=\pm ne \) with \( e=1.6\times10^{-19}\ \text{C} \), and does not depend on the speed of the body (it is relativistically invariant, unlike mass). Like charges repel and unlike charges attract. A charge needs mass to exist. A charge at rest produces only an electric field, a charge in uniform motion also produces a magnetic field, and an accelerated charge radiates electromagnetic waves.
A body can be charged by friction, conduction, induction, thermionic emission, the photoelectric effect or field emission. Conduction gives the body a charge of the same sign as the source; induction gives the opposite sign without contact.

Coulomb's Law and Superposition

The force between two point charges at rest is \( F=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2} \) along the line joining them, with \( \frac{1}{4\pi\varepsilon_0}=9\times10^9\ \text{N m}^2\text{C}^{-2} \). In vector form the force on charge 2 is \( \vec F_{21}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^3}\vec r_{12} \), where \( \vec r_{12} \) points from charge 1 to charge 2 (positive product means repulsion).
In a medium of dielectric constant K the force is divided by K. The net force on a charge from several charges is the vector sum of the individual forces (principle of superposition). Identical conducting spheres in contact share their total charge equally.

Electric Field and Lines of Force

The electric field at a point is the force per unit positive test charge, \( \vec E=\vec F/q_0 \), in N/C or V/m. A point charge gives \( E=\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2} \) directed away from a positive charge. For several sources the fields add as vectors; a charge q placed in a field E feels the force \( q\vec E \).
Lines of force start on positive charges and end on negative charges, never cross, are never closed, are perpendicular to the surface of a conductor and to equipotential surfaces, and are closer together where the field is stronger. In a uniform field they are parallel and equally spaced.

Fields of Common Charge Distributions

Ring (charge q, radius R) on its axis at distance x: \( E=\frac{kqx}{(R^2+x^2)^{3/2}} \), which is zero at the centre and has its maximum value \( \frac{2kq}{3\sqrt3R^2} \) at \( x=R/\sqrt2 \).
Infinite line charge of density λ: \( E=\frac{\lambda}{2\pi\varepsilon_0r} \). Infinite non-conducting sheet of density σ: \( E=\frac{\sigma}{2\varepsilon_0} \) on either side, independent of distance. Conducting surface: \( E=\frac{\sigma}{\varepsilon_0} \) just outside.
Uniformly charged shell (radius R, charge Q): E = 0 inside, \( E=\frac{kQ}{r^2} \) outside, and \( \frac{kQ}{R^2} \) at the surface. Uniformly charged solid sphere: \( E=\frac{kQr}{R^3} \) inside (rising linearly), \( \frac{kQ}{r^2} \) outside, with the maximum at r = R.

Gauss's Law

The flux through any closed surface is \( \oint\vec E\cdot d\vec A=\frac{q_{enc}}{\varepsilon_0} \). It is most useful for spherical, cylindrical and planar symmetry. Only charges inside the surface contribute to the flux, but the field at the surface is due to all charges. Flux through one face of a cube with a charge at its centre is \( \frac{q}{6\varepsilon_0} \).

Electric Dipole

A dipole is a pair of equal and opposite charges ±q a small distance 2a apart; its moment is \( \vec p=q(2\vec a) \), directed from −q to +q.
Axial point (distance r from the centre): \( E=\frac{2kp}{r^3} \) along p. Equatorial point: \( E=\frac{kp}{r^3} \) opposite to p. At a general point (r, θ): \( E_r=\frac{2kp\cos\theta}{r^3},\ E_\theta=\frac{kp\sin\theta}{r^3} \).
In a uniform field: torque \( \vec\tau=\vec p\times\vec E \) (maximum pE at 90°), potential energy \( U=-\vec p\cdot\vec E \) (minimum at 0°, maximum at 180°), and net force zero. In a non-uniform field there is a net force, \( F=p\frac{dE}{dx} \) along the direction of increasing field for a dipole aligned with it. The force between two short dipoles varies as 1/r⁴.

Conductors in Electrostatic Equilibrium

Inside a conductor the field is zero and the whole conductor is at one potential. Excess charge lies only on the outer surface, and the surface field is perpendicular to the surface with magnitude σ/ε₀. A cavity with no charge inside has no field; a closed conductor screens its interior from outside fields (electrostatic shielding), and a charge inside a cavity induces an equal and opposite charge on the cavity wall and an equal charge on the outer surface.
For nested shells, the charge on the inner surface of a shell is minus the charge enclosed by it; the outer surface carries the remainder. Connecting conductors with a wire makes them one conductor, so charge moves to the outer surface of the outermost one. A neutral conductor is attracted to a charged body because the nearer induced charge is stronger.

Worked Examples

Illustration 1. Find the force between charges of 3 μC and 6 μC kept 0.3 m apart in air.
Answer:
F = 9 × 10⁹ × (3 × 10⁻⁶)(6 × 10⁻⁶) / (0.3)² = 9 × 10⁹ × 18 × 10⁻¹² / 0.09 = 1.8 N (repulsive).

Illustration 2. What is the electric field 0.2 m from an infinite line of charge with density 4 μC/m?
Answer:
\( E=\frac{2k\lambda}{r}=\frac{2\times9\times10^9\times4\times10^{-6}}{0.2}=3.6\times10^{5}\ \text{N/C} \).

Illustration 3. A dipole of moment 3 × 10⁻⁸ C·m is placed in a field of 2 × 10⁴ N/C. Find the maximum torque and the work needed to turn it from the aligned position through 90°.
Answer:
τ_max = pE = 3 × 10⁻⁸ × 2 × 10⁴ = 6 × 10⁻⁴ N·m.
W = pE(cos 0° − cos 90°) = 6 × 10⁻⁴ J.