JEE Physics Electric Charges and Fields Notes Set 02

Electric Charge and Coulomb's Law

Charge is a basic property of matter. It comes in two kinds, positive and negative. Like charges repel and unlike charges attract.
Charge is conserved: it can move from one body to another but the total in an isolated system never changes.
Charge is quantised: any charge is a whole-number multiple of the electron charge, \( q = ne \) with \( e = 1.6 \times 10^{-19}\ \text{C} \).
Charge is a scalar and is additive, so the net charge on a body is the algebraic sum of its charges. A conductor lets charge move freely; an insulator does not.

Coulomb's law: the force between two point charges at rest is
\[ F = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2} = k\frac{q_1 q_2}{r^2},\quad k = 9 \times 10^{9}\ \text{N m}^2\text{C}^{-2} \]
The force acts along the line joining the charges. In a medium of dielectric constant \( K \) the force becomes \( F/K \). Forces from several charges add as vectors (superposition), and the force between any two charges is not changed by the presence of others.

Useful points:
1. Identical conducting spheres share their total charge equally when touched together.
2. For two charges with a fixed sum, the repulsion is largest when the charges are equal.
3. A third charge is in equilibrium between two like charges at the point where the two forces balance: the point is nearer the smaller charge.
4. The electrostatic force is far larger than gravitation between charged particles. For two electrons the ratio is about \( 4 \times 10^{42} \).

Electric Field

The field at a point is the force per unit positive test charge, \( \vec{E} = \vec{F}/q_0 \), with unit N/C or V/m. For a point charge \( E = kQ/r^2 \), directed away from a positive charge and towards a negative charge.
Field of common distributions:
1. Ring of charge \( Q \), radius \( R \), on its axis at distance \( x \): \( E = kQx/(R^2 + x^2)^{3/2} \). It is zero at the centre.
2. Infinite line of charge density \( \lambda \): \( E = 2k\lambda/r \).
3. Infinite sheet of surface density \( \sigma \): \( E = \sigma/2\varepsilon_0 \), independent of distance.
4. Surface of a conductor: \( E = \sigma/\varepsilon_0 \) normal to the surface; the field inside is zero.
5. Semicircular arc of radius \( R \) at its centre: \( E = 2k\lambda/R \).
A charge \( q \) in a field \( E \) feels force \( qE \) and has acceleration \( qE/m \). Field lines start on positive charges, end on negative charges, never cross, and are closer where the field is stronger.

Electric Flux and Gauss's Law

Flux through a small area is \( d\phi = \vec{E}\cdot d\vec{S} \). Gauss's law says the flux through any closed surface equals the enclosed charge divided by \( \varepsilon_0 \):
\[ \oint \vec{E}\cdot d\vec{S} = \frac{q_{enc}}{\varepsilon_0} \]
The law is used with a surface chosen to match the symmetry:
1. Charge at the centre of a cube: flux through one face is \( q/6\varepsilon_0 \).
2. Uniformly charged solid sphere: \( E = kQr/R^3 \) for \( r \le R \) and \( E = kQ/r^2 \) for \( r \ge R \).
3. Thin spherical shell: \( E = 0 \) inside and \( kQ/r^2 \) outside.
Net flux depends only on the enclosed charge, but the field at each point of the surface depends on all charges, inside and outside.

Electric Potential and Potential Energy

Potential at a point is the work done per unit charge in bringing a positive test charge from infinity to that point, \( V = W/q \) (unit volt). For a point charge \( V = kQ/r \); for several charges the potentials add as scalars.
Potential difference and field are related by \( \vec{E} = -\nabla V \), i.e. \( E = -dV/dx \) in one dimension. In a uniform field \( V = Ed \) along the field and potential falls in the direction of the field. Equipotential surfaces are perpendicular to the field lines, and no work is done in moving a charge along them.
Potential energy of two charges: \( U = kq_1q_2/r \). For a system of charges add the energy of every pair. The work done by the field in moving a charge between two points is \( q(V_A - V_B) \) and is independent of the path.
Potentials worth remembering:
1. Ring at its centre: \( V = kQ/R \).
2. Conducting sphere or shell: \( V = kQ/R \) at every point inside and on the surface, \( kQ/r \) outside.
3. On the perpendicular bisector of a dipole: \( V = 0 \).
4. Energy gained by a charge \( q \) through potential difference \( V \): \( qV \); for an electron, 1 eV = \( 1.6 \times 10^{-19} \) J.

Electric Dipole

A dipole is a pair of equal and opposite charges \( \pm q \) separated by \( 2a \) (or \( d \)). Its moment \( \vec{p} = q\vec{d} \) points from the negative to the positive charge.
1. Axial point at distance \( r \): \( E = 2kp/r^3 \) (along \( \vec{p} \)).
2. Equatorial point: \( E = kp/r^3 \) (opposite to \( \vec{p} \)).
3. Potential at angle \( \theta \) to the axis: \( V = kp\cos\theta/r^2 \).
4. In a uniform field the net force is zero and the torque is \( \tau = pE\sin\theta \), maximum at 90 degrees. The potential energy is \( U = -pE\cos\theta \).
5. For small angles the dipole oscillates with \( T = 2\pi\sqrt{I/pE} \).
6. In a non-uniform field there is also a net force. The far field of a dipole falls as \( 1/r^3 \).

Conductors in Electrostatic Equilibrium

1. The electric field inside a conductor is zero.
2. The whole conductor is an equipotential and the field at its surface is perpendicular to the surface.
3. All excess charge resides on the outer surface. Charge density is larger where the surface is more sharply curved.
4. A cavity inside a conductor is shielded from outside fields.
5. The charge induced on a conductor in a field does not depend on how well it conducts, only on its shape and the field.
6. Two concentric shells: charge placed on the outer shell changes the potential of both by the same amount, so the potential difference between them is unchanged.

Solved Examples

Example 1. Two charges of \( +3\ \mu\text{C} \) and \( +3\ \mu\text{C} \) are 10 cm apart in air. Find the force between them.
Answer:
\( F = kq_1q_2/r^2 = 9\times10^9 \times (3\times10^{-6})^2/(0.1)^2 = 8.1\ \text{N} \).
The force is repulsive.

Example 2. Find the electric field at 30 cm from a point charge of 5 nC.
Answer:
\( E = kQ/r^2 = 9\times10^9\times 5\times10^{-9}/(0.3)^2 = 500\ \text{N/C} \) directed away from the charge.

Example 3. A ring of radius 0.3 m carries a charge of 2 µC. Find the field on its axis at 0.4 m from the centre.
Answer:
\( E = kQx/(R^2+x^2)^{3/2} \), with \( R^2 + x^2 = 0.25 \) so \( (R^2+x^2)^{3/2} = 0.125 \).
\( E = 9\times10^9\times2\times10^{-6}\times0.4/0.125 = 57600\ \text{N/C} \).

Example 4. Three charges of 1 µC each are placed at the corners of an equilateral triangle of side 10 cm. Find the potential energy of the system.
Answer:
Three pairs, each at distance 0.1 m: \( U = 3kq^2/a = 3\times9\times10^9\times10^{-12}/0.1 = 0.27\ \text{J} \).

Example 5. A dipole of moment \( 3\times10^{-8} \) C m is placed in a uniform field of \( 10^5 \) N/C. Find the maximum torque and the work done in turning it from 0 to 180 degrees.
Answer:
\( \tau_{max} = pE = 0.003\ \text{N m} \).
\( W = U_{180} - U_{0} = pE - (-pE) = 2pE = 0.006\ \text{J} \).

Example 6. Find the acceleration of an electron in a uniform field of 1000 N/C.
Answer:
\( a = eE/m = 1.6\times10^{-19}\times1000/(9.1\times10^{-31}) = 1.76e+14\ \text{m/s}^2 \), directed opposite to the field.