Electric Charges & Electrostatics - Class 11 Physics

Electric Charges

Electrostatics, the earliest discovered branch of electricity, involves electric charges, the forces between them, and their behavior in materials. Just as masses are responsible for gravitational forces, electric charges are the fundamental property responsible for electrical forces.

Matter is made up of atoms. Inside every atom, positive and negative charges are held together by the enormous attraction of the electrical force. Protons carry a positive charge, while electrons carry a negative charge. Protons are bound tightly inside the nucleus of an atom and are not mobile, unlike the tinier electrons (electrons do all sorts of useful things!).

Particles with the same sign of electrical charge repel each other, and particles with opposite signs attract each other.

Crucial Question: Why don't the protons in the nucleus mutually repel and fly apart?

Answer: In addition to the repulsive electrical forces inside the nucleus, even stronger non-electrical strong nuclear forces exist. These hold the protons and neutrons together, easily overcoming the electrical repulsion over short nuclear distances.

Conservation of Charge

Conservation of charge states that: "Whenever something is charged, no electrons are created or destroyed." Electrons are simply transferred from one material to another.

For example, if you rub a glass rod with silk, a positive charge appears on the rod, and a negative charge of equal magnitude appears on the silk. Rubbing does not manufacture charge; it merely redistributes it.

An elegant example of this principle occurs in the radioactive decay of heavy nuclei:

\[^{238}\text{U} \rightarrow ^{234}\text{Th} + ^{4}\text{He}\]

The parent Uranium nucleus has $92$ protons ($+92e$). After decay, the Thorium nucleus ($90$ protons, $+90e$) and the emitted alpha particle ($2$ protons, $+2e$) yield a combined total charge of $+92e$. Thus, net charge remains conserved.

Quantization of Charge

Electric charge is not a continuous fluid. It is always composed of integral multiples of a certain minimum, elementary packet of charge.

Any net charge $q$ can be written mathematically as:

\[q = ne\]

Where $n = \pm 1, \pm 2, \pm 3, \dots$ and $e$ is the elementary elementary charge unit:

\[e = 1.6 \times 10^{-19} \text{ C}\]

Methods of Charging

Charging is effectively the intentional migration of electrons from one domain to another. We primarily achieve this through three methods: Friction, Conduction, and Induction.

1. Charging by Friction

When two distinct materials are rubbed together, thermal energy helps liberate weakly bound valence electrons from one material and deposit them onto the other. For instance, a plastic comb run through dry hair pulls electrons away from the hair, leaving the comb negatively charged.

Think About It: A plastic comb run through dry hair easily attracts small bits of paper. What happens if the hair is wet or if it is raining?

2. Charging by Induction

Charging by induction is a method used to charge a conductor without making physical contact with the charging body.

Bring a negatively charged plastic rod close to an isolated neutral metal sphere. The free electrons in the sphere are repelled to the far side, leaving the near side positively charged.
Connect the far side of the sphere to the ground (earthing) via a conducting wire. The repelled electrons escape into the earth.
Disconnect the ground wire while holding the charged rod in position.
Remove the plastic rod. The trapped positive charges redistribute uniformly across the sphere.

The Magnitude of One Coulomb

The SI unit of electric charge is the Coulomb (C). Using the quantization formula $q = ne$, we can find out how many electrons make up $1\text{ C}$:

\[1\text{ C} = n \times (1.6 \times 10^{-19}\text{ C})\] \[n = \frac{1}{1.6 \times 10^{-19}} = 6.25 \times 10^{18} \text{ electrons}\]

Because $1\text{ C}$ represents an incredibly massive cluster of charges ($6.25$ billion billion electrons), everyday laboratory physics scales down to micro-coulombs ($\mu\text{C} = 10^{-6}\text{ C}$) or nano-coulombs ($\text{nC} = 10^{-9}\text{ C}$).

Coulomb's Law

Coulomb's law states that: "The electrostatic force between any two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between their centers."

Mathematically, for two point charges $q_1$ and $q_2$ separated by distance $r$:

\[F = k\frac{q_1 q_2}{r^2}\]

Where $k$ is the electrostatic force constant. In a vacuum or air medium:

\[k = \frac{1}{4\pi\varepsilon_0} \approx 9 \times 10^9 \text{ N}\cdot\text{m}^2\text{C}^{-2}\]

Where $\varepsilon_0 = 8.854 \times 10^{-12} \text{ C}^2\text{N}^{-1}\text{m}^{-2}$ is the absolute permittivity of free space.

Relative Permittivity ($\varepsilon_r$)

When charges are placed inside a material medium (like water or glass), the electrostatic force decreases. Relative permittivity (or dielectric constant) is defined as:

\[\varepsilon_r = \frac{\varepsilon}{\varepsilon_0} = \frac{F_{\text{vacuum}}}{F_{\text{medium}}}\]

Conceptual & Numerical Challenges

1. Compare the electrostatic force between two points in air and water if the permittivity of water is 80 times the absolute permittivity of air/vacuum.

2. Compare the strength of gravitational and electrostatic force between a proton and an electron separated by a unit distance in air. Comment on the result.

3. Prove that the electrostatic force between two charged particles separated such that a distance $r_1$ is filled with air and distance $r_2$ is filled with a medium of dielectric constant $\varepsilon_r$ is given by: \[F = \frac{q_1 q_2}{4\pi \varepsilon_0 (r_1 + \sqrt{\varepsilon_r} r_2)^2}\]

4. Two particles A and B having charges $8.0 \times 10^{-6}\text{ C}$ and $-2.0 \times 10^{-6}\text{ C}$ respectively are held fixed with a separation of $20\text{ cm}$. Where should a third charged particle be placed so that it does not experience a net electric force?

5. A charge $Q$ is to be divided on two objects. What should be the values of the charges on the objects so that the force between the objects can be maximum?

6. Two particles, each having a mass of $5\text{ g}$ and charge $1.0 \times 10^{-7}\text{ C}$, stay in limiting equilibrium on a horizontal table with a separation of $10\text{ cm}$ between them. Find the coefficient of static friction between each particle and the table.

Physics in Depth