Quantization

The quantisation of charge is a fundamental principle in physics which states that electric charge exists in discrete, indivisible units rather than being continuous. According to this concept, every observable charge is an integer multiple of a basic unit of charge, denoted by e, where$[e=1.6\times 10^{-19}\text{C}]$

This means that all charged particles, such as electrons and protons, carry charges that are exact multiples of this elementary charge. No particle has been found to possess a fraction of this value under normal conditions. The discovery of quantised charge has deepened our understanding of atomic structure and electric phenomena, revealing that charge is conserved and transferred in fixed packets rather than varying smoothly.

1.0Electric Charge

Electric charge is a fundamental property of elementary particles—such as electrons and protons—from which all matter is formed. This property is responsible for the electric forces of attraction or repulsion that objects exert on one another.

Cause of Electric Forces

• The presence of electric charge on objects leads to electric interactions. Like charges repel each other, while unlike charges attract, giving rise to various electrical phenomena.

Intrinsic Nature

• Electric charge is an inherent and unchangeable characteristic of particles. Electrons always carry a negative charge, and protons always carry a positive charge, regardless of their environment.
• Electric charge is a scalar physical quantity, meaning it has magnitude only and no direction.
• The standard unit for measuring electric charge is the coulomb (C).

Elementary Charge: The smallest unit of charge that can exist independently is called the elementary charge, denoted by e.

• Charge of a proton: ( +e )
• Charge of an electron: ( -e )
• Value of elementary charge: $[e=1.6\times 10^{-19}\text{C}]$

Electrical Neutrality: A material body is electrically neutral when it contains an equal number of protons and electrons. In this state, positive and negative charges balance each other.

Positive and Negative Charging:

• If a body gains extra electrons, it becomes negatively charged.
• If a body loses electrons or has an excess of protons, it becomes positively charged.

Charge Conservation:

Electric charge can neither be created nor destroyed. It can only be transferred from one body to another, ensuring the total charge in an isolated system remains constant.

2.0Quantisation of Charge:

Meaning of Quantisation: Quantisation of a physical quantity means that the quantity cannot take any arbitrary or continuous value. Instead, it can change only in fixed steps or discrete units.

Discrete vs Continuous Values: A quantised quantity has specific allowed values, similar to how a building has only definite floors (ground floor, first floor, second floor, etc.) and cannot have a floor in between.

• Examples of Quantisation
  1. Energy levels of an electron in an atom are quantised.
  2. Electric charge is also quantised; it changes only in discrete units.

Quantum of a Physical Quantity: The smallest possible change or the minimum unit by which a physical quantity can vary is called its quantum.

Experimental Observation: Experiments show that the electric charge of any object—large or small—is always an integer multiple of a minimum, basic unit of charge.

Elementary Charge: This minimum charge is the charge on an electron or proton, known as the elementary charge (e):$[e=1.6\times 10^{-19}\text{C}]$

Charge on an electron: ( -e )
Charge on a proton: ( +e )
Charge on an alpha particle: ( +2e )

More Accurate Value: A more precise modern value of the elementary charge is $[e=1.602192\times 10^{-19}\text{C}]$

Reason for Quantisation of Charge: During processes such as rubbing or charging, only an integral number of electrons can be transferred between bodies. Since electrons carry a fixed charge, the total charge must always be an integer multiple of e.

3.0Additional Points on Charge Quantisation

Quark Model Insight: High-energy physics has shown that protons and neutrons are made of quarks, which carry fractional charges:

• Up quark: $[+\frac{2}{3}e]$
• Down quark: $[-\frac{1}{3}e]$

Even if quarks are confirmed as fundamental particles, quantisation still holds; only the base unit of charge would then become$[\frac{e}{3}]$

Universality of Quantisation: Quantisation is a universal principle in nature. Not only charge, but energy and angular momentum of electrons are also quantised. (However, quantisation of mass is still not established.)

Total charge ( q ) on a body is given by $q=\pm ne$

Where ( n ) is an integer, $[n=0,1,2,3,\dots ]$ and e is the elementary charge

Note: Charge can only increase or decrease in whole-number multiples of e; fractional values like ( 0.5e ) are never observed in macroscopic objects.

4.0Experimental Verification of Charge Quantisation

1. Faraday’s Laws of Electrolysis :Faraday’s experiments were the first to indicate that electric charge exists in discrete packets.
2. Millikan’s Oil Drop Experiment : Millikan measured the charge on individual oil droplets and found that all charges were integer multiples of a fundamental value, confirming that charge is quantised.

Conditions Under Which Charge Quantisation Is Ignored

• Macroscopic Situations: In everyday life, the quantisation of charge is usually ignored because the elementary charge eee is extremely small, while the number of electrons involved is extremely large.
• Continuity Approximation: In electrical circuits or appliances, charge appears to vary continuously. For example, when a 60-W bulb is switched on, around 2 1018 electrons pass through its filament every second. The individual electron charges are too tiny to cause noticeable "jumps."
• Relevance at Microscopic Scale: Quantisation becomes important only when dealing with very small charges—of the order of a few tens or hundreds of electrons.

Neutral Matter

• Condition for Neutrality: A body is electrically neutral when it contains equal numbers of protons and electrons.

Charging a Body

• Excess electrons → negatively charged body
• Deficit of electrons (or excess protons) → positively charged body

5.0Millikan’s Oil Drop Experiment

Millikan’s Oil Drop Experiment measured the charge of an electron by observing the motion of tiny charged oil droplets in an electric field. By adjusting the electric field strength so that a droplet was held in equilibrium, Millikan calculated its charge, which always appeared as an integral multiple of the elementary charge e.

The experiment used two parallel metal plates separated by an insulating rod. The top plate had holes for light to pass through and for viewing with a microscope. Special low–vapour-pressure oil was used to prevent evaporation during observation.

Procedure

• Oil droplets were sprayed using an atomizer into the chamber.
• The droplets’ downward motion was observed to determine their mass and terminal velocity.
• X-rays ionized the air, allowing droplets to pick up electric charge.
• An electric field was applied between the plates; it exerted an upward force on the charged droplet.
• The field strength was adjusted until the droplet was in equilibrium, balancing gravitational force.

At equilibrium: QE = mg

where Q is the charge on the droplet, E is electric field strength, m is the droplet’s mass, and g is gravitational acceleration.

Millikan found that every droplet carried charge in discrete amounts, always an integer multiple of $e=1.6\times 10^{-19}\text{C}$.

Illustration-1

In a modified version of Millikan’s experiment, an oil droplet of radius $r=1.5\times 10^{-6}m$ and density $\rho =900kg/m^3$ is suspended between two horizontal parallel plates separated by a distance of d=5 mm by applying a potential difference of V=400 V. Air’s viscosity and buoyancy are negligible. Assuming the droplet carries n electrons, find the number of electrons (n) that must be present on the droplet so that it remains in equilibrium.

Solution:

Electric field between the plates:

$E=\frac{V}{d}=\frac{400}{5\times 10^{-3}}=8\times 10^{4}V/m$

Mass of the oil droplet,

Volume of droplet: $V=\frac{4}{3}\pi r^3$

$V=\frac{4}{3}\pi (1.5\times 10^{-6}{)}^3=1.41\times 10^{-17}{\text{m}}^3$

Mass:

$m=\rho V=900\times 1.41\times 10^{-17}=1.27\times 10^{-14}\text{kg}$

Gravitational force on the droplet

$F_g=mg=1.27\times 10^{-14}\times 9.8=1.24\times 10^{-13}\text{N}$

Electric force required for equilibrium, $F_e=F_g\Rightarrow qE=mg$

Charge on droplet: $q=\frac{mg}{E}=\frac{1.24\times 10^{-13}}{8\times 10^4}=1.55\times 10^{-18}\text{C}$

Number of electrons on the droplet:$n=\frac{q}{e}=\frac{1.55\times 10^{-18}}{1.6\times 10^{-19}}=9.7$

Since charge must be an integral multiple of the elementary charge, $n=10\text{electrons}$