CBSE Class 12 Physics

Electric charge

Matter carries electric charge of two kinds, called positive and negative. Charges of the same kind repel and charges of different kinds attract. A body is charged when it has an excess of one kind, and it is neutral when the positive and negative charges in it balance exactly. Rubbing two different insulators together moves electrons from one to the other, so one ends up negative and the other positive by the same amount; nothing is created, only transferred.

The SI unit of charge is the coulomb (C). The smallest free charge found in nature is the magnitude of the charge on an electron, the elementary charge $e$.

Quantisation of charge

The charge on any body is a whole-number multiple of $e$.

Because $e$ is so small, a charge of even a microcoulomb contains about $6\times10^{12}$ elementary charges, so charge on everyday bodies looks continuous and the steps are invisible. Quantisation matters for atoms and for very small charges.

Conservation of charge

The total charge of an isolated system does not change. Charges can be moved from one part to another, and pairs of equal and opposite charges can appear or vanish together, but the algebraic sum stays the same.

Additivity of charge

Charge is a scalar and it adds algebraically, with signs. A system holding charges $+5\ \mu\mathrm{C}$, $-3\ \mu\mathrm{C}$ and $+1\ \mu\mathrm{C}$ has a total charge of $+3\ \mu\mathrm{C}$.

Conductors and insulators

In a conductor some charges (the free electrons in a metal) can move through the material, so charge given to one place spreads over the whole body. In an insulator the charges are bound to their atoms and stay where they are put. Charging by induction uses this: a charged rod brought near an isolated conducting sphere pulls charge of the opposite sign to the near side and pushes the same sign to the far side; earthing the far side removes the repelled charge, and taking the rod away leaves the sphere with a net charge opposite in sign to the rod.

Coulomb's law

The force between two point charges at rest acts along the line joining them. Its size is proportional to the product of the charges and inversely proportional to the square of the distance between them. A point charge is a charged body so small, compared with the distance to other charges, that its size does not matter.

$F$ is in newtons, $q_1$ and $q_2$ in coulombs and $r$ in metres. The constant $\varepsilon_0$ is the permittivity of free space.

The force is repulsive when $q_1q_2$ is positive and attractive when it is negative. Coulomb's law is an inverse-square law, so doubling the distance cuts the force to a quarter, and it obeys Newton's third law: each charge pulls or pushes the other with a force of the same size in the opposite direction.

If the charges are in a medium rather than a vacuum, the force is smaller. The medium's relative permittivity (dielectric constant) $\varepsilon_r$ gives $F = \dfrac{1}{4\pi\varepsilon_0\varepsilon_r}\dfrac{q_1q_2}{r^2}$, so the force in the medium is $1/\varepsilon_r$ times the force in vacuum at the same separation.

Vector form

For charge $q_1$ at position $\vec r_1$ and $q_2$ at $\vec r_2$, the force on $q_2$ due to $q_1$ is

where $\hat r_{21}$ is the unit vector pointing from $q_1$ to $q_2$ and $r_{21}$ is the distance between them. With this form the sign takes care of itself: if $q_1q_2>0$ the force on $q_2$ points away from $q_1$, and if it is negative the force points towards it.

Worked example: two charges

Two small spheres carry $+2.0\ \mu\mathrm{C}$ and $-3.0\ \mu\mathrm{C}$ and are $0.30\ \mathrm{m}$ apart in air.

The force is attractive, and each sphere feels $0.60\ \mathrm{N}$ directed towards the other.

Comparing electric and gravitational forces

Between a proton and an electron in a hydrogen atom, about $5.3\times10^{-11}\ \mathrm{m}$ apart, the electric force is about $8\times10^{-8}\ \mathrm{N}$, while the gravitational force is about $3.6\times10^{-47}\ \mathrm{N}$. The ratio is of the order of $10^{39}$. Gravity is the weaker force by an enormous factor, and it only dominates on large scales because matter as a whole is electrically neutral, so the electric forces cancel.

Superposition principle

The force between two charges is not affected by the presence of other charges. So the total force on any charge is the vector sum of the forces that each of the other charges would exert on it alone.

Worked example: three charges in a line

Charges $q_A = +4.0\ \mu\mathrm{C}$, $q_B = -2.0\ \mu\mathrm{C}$ and $q_C = +1.0\ \mu\mathrm{C}$ lie on a line, with $B$ at $0.10\ \mathrm{m}$ from $A$ and $C$ at $0.10\ \mathrm{m}$ beyond $B$. Find the force on $B$.

The force from $A$ on $B$ is attractive, towards $A$:

The force from $C$ on $B$ is also attractive, towards $C$, and in the opposite direction:

The net force is $7.2 - 1.8 = 5.4\ \mathrm{N}$ towards $A$.

When the charges are not on one line, each force is resolved into components along two perpendicular axes, the components are added separately, and the magnitude and direction of the resultant are found from them.

Continuous charge distribution

When a body holds a very large number of tiny charges close together, it is easier to describe it by a charge density than to list every charge. The density is defined so that the total charge in a small element is the density times the size of the element.

The force or field of such a body is found by dividing it into small elements, treating each element as a point charge $\mathrm{d}q$ (equal to $\lambda\,\mathrm{d}l$, $\sigma\,\mathrm{d}S$ or $\rho\,\mathrm{d}V$), and adding the contributions, which in the limit becomes an integral. For a uniformly charged ring of radius $R$ and total charge $Q$, the charge on each short arc is $\mathrm{d}q = \lambda\,\mathrm{d}l$ with $\lambda = Q/2\pi R$.