Class 12 Physics: Electrostatics, Charge Properties, and Charging Methods

Introduction to Class 12 Physics & Electromagnetism Overview

Starting Class 12th Physics involves establishing a baseline of $T = 0$. Past performance, neglect, or weak foundations in Class 11th are set aside to focus entirely on the new syllabus.

Electromagnetism is the dominant branch of physics in Class 12th, covering a major portion of the entire curriculum. Electromagnetism is divided into two primary sub-disciplines:

1. Electrostatics: The study of electric charges at rest ($\text{charge at rest}$). This sub-discipline covers:

   • Electric charges and their fundamental properties

   • Electric forces and Coulomb's Law

   • Electric field and electric potential

   • Gauss's Law

   • Capacitors and dielectric media

2. Electrodynamics: The study of electric charges in motion ($\text{charge in motion}$). This dynamic branch encompasses:

   • Electric current (movement of charge carriers)

   • Magnetism and magnetic effects of moving charges

   • Electromagnetic Induction (EMI)

   • Alternating Current (AC)

Following Electromagnetism, the Class 12 physics curriculum proceeds to Wave Optics, Ray Optics, and Modern Physics.

Fundamental Concepts of Electric Charge

Electric charge is an intrinsic property of matter. An intrinsic property represents an ultimate baseline attribute of a physical entity that cannot be broken down or explained by deeper underlying structures. Mass is similarly an intrinsic fundamental property of matter.

Historical records indicate that electric charge phenomena were first documented in Greek mythology around $600\,\text{BC}$, when rubbing certain substances together caused them to attract lightweight objects.

• Symbol: Charge is represented by the symbol $Q$ or $q$.

• SI Unit: The Standard International unit of charge is the Coulomb, denoted by the symbol $C$.

• Physical Quantity: Charge is a scalar quantity, possessing magnitude only and no spatial direction.

• Types of Charge: There are two distinct types of electric charge:

  • Positive charge ($+$)

  • Negative charge ($–$)

Electrostatic force interactions follow a fundamental rule: like charges repel each other ($+$ repels $+$, and $-$ repels $-$), whereas unlike charges attract each other ($+$ and $-$ attract).

Key Differences Between Charge and Mass

Charge and mass are both fundamental intrinsic properties of matter, but they possess crucial distinctions. Charge always exists alongside mass ($\text{charge always comes with mass}$). Any physical entity carrying an electric charge must possess a non-zero rest mass, and transferring charge between bodies inherently transfers mass. However, the converse is not required: a body with mass does not necessarily carry a net electric charge.

Differences between Charge and Mass include:

1. Sign and Types:

   • Charge can be positive, negative, or zero.

   • Mass is strictly a positive scalar quantity ($+$). A negative mass (e.g., $-5\,\text{kg}$) does not exist.

2. Speed Dependency (Relativistic Invariance):

   • Mass depends on speed: The relativistic mass $m$ of an object increases with its velocity $v$ according to the relation:      $m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$      Where:

     • $m_0$ = Rest mass of the object (mass when $v = 0$)

     • $m$ = Relativistic mass of the moving object

     • $v$ = Velocity of the object

     • $c$ = Speed of light in vacuum ($c \approx 3 \times 10^8\,\text{m/s}$)

   Because the speed of any physical object $v < c$, the ratio $\frac{v^2}{c^2} < 1$, which means $\sqrt{1 - \frac{v^2}{c^2}} < 1$. Dividing the rest mass $m_0$ by a value less than $1$ yields $m > m_0$.

   In classical macroscopic mechanics (Class 11 topics such as blocks, pulleys, and pendulums), typical speeds are very small compared to the speed of light ($v \ll c$, e.g., $50\,\text{m/s}$, $100\,\text{m/s}$, or $200\,\text{m/s}$). Consequently, $\frac{v^2}{c^2} \approx 0$, $\sqrt{1 - 0} = 1$, and $m \approx m_0$. Thus, relativistic mass variation is negligible in everyday classical scenarios.

   In quantum electrodynamics and subatomic physics, particles such as electrons move at speeds around $10^6\,\text{m/s}$, beta particles reach speeds up to $10^7\,\text{m/s}$, and electromagnetic waves travel at $c = 3 \times 10^8\,\text{m/s}$. At these high speeds, relativistic mass variations become significant.

• Charge is independent of speed: The magnitude of charge on a particle remains strictly constant regardless of its velocity ($\text{charge is invariant under motion}$). Speed affects a subatomic particle's mass, but leaves its electric charge unchanged.

Principle of Conservation of Charge

The total electric charge of an isolated system remains constant over time. An isolated system is a system enclosed within non-conducting or insulating boundaries that prevent any charge transfer across its borders.

While popular formulations state that charge can neither be created nor destroyed but only transferred from one form or body to another, the precise statement is that the net charge of an isolated system is conserved.

Mathematical System Demonstration

Consider an isolated system containing three distinct bodies with initial charges:

• Body 1: $+3\,\text{C}$

• Body 2: $+1\,\text{C}$

• Body 3: $-2\,\text{C}$

The total initial charge of the isolated system is: $Q_{\text{initial}} = (+3\,\text{C}) + (+1\,\text{C}) + (-2\,\text{C}) = +2\,\text{C}$

If the system is shaken internally and allowed to redistribute charge, and subsequent measurements reveal:

• Body 1: $+2\,\text{C}$

• Body 2: $+3\,\text{C}$

• Body 3: $Q$

Applying charge conservation ($Q_{\text{initial}} = Q_{\text{final}}$): $+2\,\text{C} = (+2\,\text{C}) + (+3\,\text{C}) + Q$ $+2 = +5 + Q$ $Q = 2 - 5 = -3\,\text{C}$

Thus, Body 3 carries a charge of $-3\,\text{C}$, maintaining the total system charge at $+2\,\text{C}$.

Applications and Empirical Basis

Conservation of charge is experimentally verified to be universally true. It has no formal mathematical derivation from lower-level principles, but no experimental violation has ever been observed.

In nuclear radioactivity equations, conservation of charge dictates the charge balance of daughter products. For instance, in the alpha decay of a Thorium nucleus ($\text{Th}$, atomic number $90$): $\text{Thorium } (+90\,e) \rightarrow \alpha\text{-particle } (+2\,e) + \text{Daughter Nucleus } (+88\,e)$

The daughter nucleus retains $+88\,e$ to conserve the net nuclear charge of $+90\,e$.

Quantization of Charge

Quantization means that a physical quantity is available only in fixed, discrete amounts rather than across a continuous range.

The smallest unit of charge that can exist independently in the universe is the magnitude of charge on a single electron (or proton), termed the fundamental charge ($e$): $e = 1.6 \times 10^{-19}\,\text{C}$

Subatomic entities such as quarks and anti-quarks possess fractional charges (e.g., $\pm \frac{1}{3}e$ or $\pm \frac{2}{3}e$), but quarks do not exist independently in nature. Therefore, $e = 1.6 \times 10^{-19}\,\text{C}$ remains the smallest free, independent charge unit.

Mathematical Formulation

Charge exists exclusively in integral multiples of the elementary charge $e$: $Q = \pm n\,e$ Where:

• $Q$ = Net charge on any physical body

• $n$ = Integer ($n = 1, 2, 3, 4, \dots$)

• $e$ = Fundamental elementary charge ($1.6 \times 10^{-19}\,\text{C}$)

Allowed charge values on any body include $\pm 1\,e$, $\pm 2\,e$, $\pm 3\,e$, $\dots$, $\pm n\,e$. Non-integer multiples such as $2.5\,e$, $\frac{3}{7}\,e$, or $-4.8\,e$ cannot exist because electrons cannot be split or transferred in fractional parts.

Sample Problem

Problem: Can a charge of $+8 \times 10^{-18}\,\text{C}$ be given to a body?

Solution: Apply the quantization equation $Q = n\,e$: $8 \times 10^{-18}\,\text{C} = n \times (1.6 \times 10^{-19}\,\text{C})$ $n = \frac{8 \times 10^{-18}}{1.6 \times 10^{-19}}$ $n = \left(\frac{8}{1.6}\right) \times 10^{-18 - (-19)}$ $n = 5 \times 10^1 = 50$

Since $n = 50$ is an integer, this charge can be imparted to a body by removing exactly $50$ electrons. If $n$ had resulted in a non-integer value (such as $49.8$), imparting that charge would be physically impossible.

Methods of Charging a Body

Neutral objects contain equal numbers of positive protons and negative electrons (e.g., Hydrogen has $1$ proton and $1$ electron; Sodium has $11$ protons and $11$ electrons). Charging involves creating an imbalance between electrons and protons.

There are three primary methods to charge a body:

1. Charging by Conduction

Charging by conduction requires direct physical contact between conductors.

• When a negatively charged conductor (possessing an excess of free electrons) directly touches an uncharged neutral conductor, electrons transfer from the charged body to the uncharged body due to electrostatic repulsion and concentration gradients.

• Equal Distribution Condition: If two spherical conductors possess identical shape, size, and material composition, the net initial charge divides equally upon contact.

  • If Body A carries initial charge $Q$ and Body B carries charge $0$, touching them yields a final charge of $\frac{Q}{2}$ on each conductor.

  • If the conductors differ in geometry, sharp edges, size, or material, the charge redistributes unequally based on local capacitance and geometry.

2. Charging by Induction

Charging by induction occurs without physical contact between the charging source and the object.

Method A: Two Conductors

1. Bring a negatively charged rod near a neutral conductor (Body A) without touching it.

2. Electrostatic repulsion drives free electrons in Body A to the far end, leaving the near end positively charged (polarization).

3. Protons do not move because they are bound in atomic nuclei and are much heavier than electrons ($m_p \approx 1.672 \times 10^{-27}\,\text{kg}$, whereas $m_e \approx 9.1 \times 10^{-31}\,\text{kg}$, making $m_p \approx 1836 \times m_e$).

4. Bring a second neutral conductor (Body B) into contact with the far side of Body A. Excess repelled electrons flow into Body B.

5. Separate Body B from Body A while keeping the external negative rod in position.

6. Remove the negative rod. Body A remains positively charged, while Body B becomes negatively charged.

Method B: Earthing / Grounding

1. To make a neutral conductor negatively charged, bring a positively charged rod near it to polarize its internal charges.

2. Ground (earth) the far end of the conductor. The Earth acts as an infinite ocean of free electrons.

3. Electrons flow from the Earth into the conductor to neutralize the positive charge at the far end.

4. Disconnect the ground connection while holding the positive rod in place.

5. Remove the positive rod. The body retains a net negative charge.

Induction Rule: The near end always acquires an opposite polarity, while the far end acquires the same polarity as the inducing charge.

3. Charging by Friction

Charging by friction is predominantly used for insulating materials.

• Rubbing two dissimilar insulating materials together converts mechanical kinetic energy into thermal energy.

• Thermal energy supplies the necessary ionization energy to outer-shell electrons in surface atoms.

• Outer-shell electrons absorb energy and transfer from the material with lower work function / electron affinity to the material with higher electron affinity.

• The material losing electrons becomes positively charged, while the material gaining electrons becomes negatively charged.

Mass Transfer Rule

Because charge transfer occurs via the movement of physical electrons (each having rest mass $m_e \approx 9.1 \times 10^{-31}\,\text{kg}$), charge transfer is always accompanied by mass transfer:

• A negatively charged body gains electrons, so its mass increases.

• A positively charged body loses electrons, so its mass decreases.

Real-World Examples

• Rubbing a plastic scale against hair causes electron transfer via friction. The charged scale then polarizes small neutral bits of paper by induction, causing them to adhere to the scale.

• Rubbing a rubber balloon against dry hair allows it to stick to a neutral wall via frictional charging followed by surface electrostatic induction.