Electric Charges and Fields Comprehensive Study Notes

Introduction to Electrostatics

• Phenomenological Observations of Static Electricity:
  • Static electricity is experienced as a spark or crackle when removing synthetic clothes/sweaters in dry weather.
  • Lightning is a large-scale example of electric discharge during thunderstorms.
  • A sensation of electric shock occurs when opening car doors or holding iron bars on buses after sliding across a seat, caused by the discharge of electric charges accumulated through the rubbing of insulating surfaces.
• Definition of Terms:
  • Static: Refers to anything that does not move or change with time.
  • Electrostatics: The branch of physics dealing with the study of forces, fields, and potentials arising from static charges.

Electric Charge and Historical Context

• History of Discovery:
  • Thales of Miletus (Greece, approx. 600 BC) discovered that amber rubbed with wool or silk attracts light objects.
  • The word "electricity" is derived from the Greek word "elektron," meaning amber.
• Experiments with Electrification:
  • Two glass rods rubbed with wool/silk repel each other.
  • The wool/silk pieces used for rubbing also repel each other, but the glass rod and wool attract.
  • Two plastic rods rubbed with cat’s fur repel each other but attract the fur.
  • A plastic rod attracts a glass rod, while the plastic rod repels the silk/wool used on the glass.
• Conclusions on Charges:
  • There are only two kinds of electric charge.
  • Bodies like glass/plastic rods, silk, and fur become electrified or charged on rubbing.
  • Like charges repel and unlike charges attract.
  • Polarity of Charge: The property that differentiates the two kinds of charges.
  • Benjamin Franklin named these two types Positive and Negative.
  • Convention: Glass rod/cat’s fur is positive; plastic rod/silk is negative.
  • Nuetralization: Unlike charges of equal magnitude nullify each other's effect when brought into contact.

The Gold-Leaf Electroscope

• Structure: Consists of a vertical metal rod housed in a box with two thin gold leaves attached at the bottom.
• Operation: When a charged object touches the top metal knob, charge flows to the leaves, causing them to diverge due to repulsion.
• Indicator: The degree of divergence indicates the amount of charge.

Microscopic Origin and Conductors/Insulators

• Origin of Charge:
  • All matter is made of atoms/molecules. Materials are naturally neutral because positive and negative charges are balanced.
  • Forces holding molecules together (adhesive forces, surface tension) are fundamentally electrical.
  • Charging involves transferring electrons (less tightly bound in solids) from one body to another.
  • Positive Charge: Result of a deficit of electrons.
  • Negative Charge: Result of an excess of electrons.
  • Conservation during Rubbing: No new charge is created; electrons are simply relocated.
• Materials Classification:
  • Conductors: Readily allow electricity to pass; contain mobile charges (free electrons). Examples: Metals, human/animal bodies, Earth.
  • Insulators: Offer high resistance to electricity. Examples: Glass, porcelain, plastic, nylon, wood.
  • Semiconductors: Intermediate resistance (discussed later).
• Grounding/Earthing: In conductors, transferred charge distributes over the surface. If a conductor (like a metal spoon) is touched by a human, charges leak through the body to the ground. Using a plastic handle prevents this leakage.

Basic Properties of Electric Charge

• Point Charges: If the sizes of charged bodies are much smaller than the distance between them, they are treated as point charges (concentrated at a single point).
• Additivity of Charges:
  • Total charge is the algebraic sum of individual charges in a system.
  • Charges are scalars; sign matters during addition.
  • For charges $q_1, q_2, ..., q_n$, total charge $Q = q_1 + q_2 + ... + q_n$.
• Conservation of Charge:
  • The total charge of an isolated system is always conserved.
  • Charges can be redistributed, but the net charge cannot be created or destroyed.
  • In a process like neutron decay ($n \rightarrow p + e$), the total charge remains zero.
• Quantisation of Charge:
  • All free charges are integral multiples of a basic unit $e$.
  • Formula: $q = ne$, where $n$ is an integer ($0, \pm 1, \pm 2, \pm 3, ...$).
  • Value of e: $e = 1.602192 \times 10^{-19}\,C$.
  • Quantisation was first suggested by Faraday and experimentally demonstrated by Millikan (1912).
  • For macroscopic charges ($\mu C$ level), quantisation is ignored because the "graininess" is not visible; it is treated as a continuous distribution.
• Unit of Charge: The Coulomb (C). In SI, 1 C is defined via current: it is the charge flowing in 1 s if the current is 1 A.
  • $1\,\mu C = 10^{-6}\,C$.
  • $1\,mC = 10^{-3}\,C$.

Numeric Examples

• Example 1.1: If $10^9$ electrons move out of a body every second, the time to accumulate 1 C is approximately 198 years ($t = \frac{1}{1.6 \times 10^{-10}}\,s$).
• Example 1.2: A cup of water ($250\,g$) contains approximately $1.34 \times 10^7\,C$ of positive and negative charge. ($N = \frac{250}{18} \times 6.02 \times 10^{23}$ molecules, each having 10 protons and 10 electrons).

Coulomb’s Law

• Definition: The electrostatic force between two point charges $q_1, q_2$ is inversely proportional to the square of the distance $r$ between them and directly proportional to the product of the magnitudes of the charges.
• Mathematical Expression:$F = k \frac{q_1q_2}{r^2}$
• Constant k: In SI units, $k \approx 9 \times 10^9\,N\,m^2\,C^{-2}$. It is often written as:     $k = \frac{1}{4\pi\epsilon_0}$
• Permittivity of Free Space ($\epsilon_0$):$\epsilon_0 = 8.854 \times 10^{-12}\,C^2\,N^{-1}\,m^{-2}$
• Vector Form:$\mathbf{F}_{21} = \frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r_{21}^2} \mathbf{\hat{r}}_{21}$
  • Force is along the line joining the charges. Like charges repel ($\mathbf{F}_{21}$ in direction of $\mathbf{\hat{r}}_{21}$); unlike charges attract ($\mathbf{F}_{21}$ in direction of $-\mathbf{\hat{r}}_{21}$).
  • Forces satisfy Newton’s Third Law: $\mathbf{F}_{12} = -\mathbf{F}_{21}$.
• Torsion Balance: The device used by Charles Augustin de Coulomb to measure these forces.

Forces Between Multiple Charges (Superposition Principle)

• Principle: The force on any charge due to multiple other charges is the vector sum of all the forces exerted on that charge by the other charges taken individually. The individual forces remain unaffected by the presence of other charges.
• Formula for Charge 1 in a system of n charges:$\mathbf{F}_1 = \mathbf{F}_{12} + \mathbf{F}_{13} + ... + \mathbf{F}_{1n} = \frac{q_1}{4\pi\epsilon_0} \sum_{i=2}^{n} \frac{q_i}{r_{1i}^2} \mathbf{\hat{r}}_{1i}$

Electric Field

• Definition: The electric field $\mathbf{E}$ at a point due to a source charge $Q$ is the force experienced by a unit positive test charge $q$ placed at that point, divided by the test charge ($\mathbf{E} = \mathbf{F}/q$).
• Field of a Point Charge:$\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \mathbf{\hat{r}}$
• Properties:
  • Unit: Newtons per Coulomb ($N\,C^{-1}$).
  • Field is independent of the test charge $q$ (for sufficiently small $q$).
  • Positive source charge: Field points radially outward.
  • Negative source charge: Field points radially inward.
  • Exhibits spherical symmetry: magnitude depends only on distance $r$.
• Physical Significance:
  • Allows handling of time-dependent electromagnetic interactions where delays occur due to the finite speed of light ($c$).
  • Fields are physical entities that can transport energy and have their own dynamics.

Electric Field Lines

• Visual Representation: Introduced by Michael Faraday.
• Definition: A curve where the tangent at any point gives the direction of the net electric field vector at that point.
• Key Properties:
  1. Field lines start at positive charges and end at negative charges (or infinity).
  2. In charge-free regions, they are continuous curves.
  3. Two field lines never cross (as this would imply two field directions at one point).
  4. They do not form closed loops (due to the conservative nature of electric fields).
• Field Strength: Indicated by the density of lines. Crowded lines indicate strong fields; spaced-apart lines indicate weak fields.

Electric Flux

• Definition: A measure of the number of field lines crossing a surface area element $\Delta S$.
• Formula:$\Phi = \mathbf{E} \cdot \Delta \mathbf{S} = E \Delta S \cos(\theta)$
  • $\theta$ is the angle between $\mathbf{E}$ and the normal to the area $\mathbf{\hat{n}}$.
• Vector Area: Area is treated as a vector directed along the outward normal to the surface.
• Unit: $N\,C^{-1}\,m^2$.

Electric Dipole

• Definition: A pair of equal and opposite charges ($q, -q$) separated by a distance $2a$.
• Dipole Moment (p): A vector directed from $-q$ to $+q$$\mathbf{p} = q(2a) \mathbf{\hat{p}}$
• Electric Field of a Dipole:
  1. On the Axis (at distance r):$\mathbf{E} = \frac{2\mathbf{p}r}{4\pi\epsilon_0(r^2 - a^2)^2}$. For $r \gg a$, $E \approx \frac{2p}{4\pi\epsilon_0 r^3}$.
  2. On the Equatorial Plane (at distance r):$\mathbf{E} = \frac{-\mathbf{p}}{4\pi\epsilon_0(r^2 + a^2)^{3/2}}$. For $r \gg a$, $E \approx \frac{-p}{4\pi\epsilon_0 r^3}$.
• Important Note: Dipole fields fall off as $1/r^3$, faster than point charges ($1/r^2$).
• Physical Significance: Polar molecules (like $H_2O$) have permanent dipole moments. Non-polar molecules (like $CO_2$) gain induced dipole moments in an external field.

Dipole in external field

• Uniform Field: Expereinces a torque $\tau = \mathbf{p} \times \mathbf{E}$ (magnitude $\tau = p E \sin(\theta)$). Net force is zero.
• Non-uniform Field: Experiences both torque and a net force. The force depends on the gradient of the field and dipole orientation.

Continuous Charge Distribution

• Linear Charge Density ($\lambda$): Charge per unit length ($C\,m^{-1}$). $\lambda = \Delta Q / \Delta l$.
• Surface Charge Density ($\sigma$): Charge per unit area ($C\,m^{-2}$). $\sigma = \Delta Q / \Delta S$.
• Volume Charge Density ($\rho$): Charge per unit volume ($C\,m^{-3}$). $\rho = \Delta Q / \Delta V$.

Gauss’s Law

• Definition: The total electric flux through any closed surface $S$ is equal to the total charge $q$ enclosed by the surface divided by the permittivity of free space $\epsilon_0$.
• Mathematical Formula:$\Phi = \sum \mathbf{E} \cdot \Delta \mathbf{S} = \frac{q_{enclosed}}{\epsilon_0}$
• Key Points:
  1. True for any closed surface of any shape/size.
  2. $q$ is the sum of all charges inside; charges outside do not contribute to the net flux (though they contributes to $\mathbf{E}$).
  3. Gaussian Surface: The chosen closed surface for calculation.
  4. Based on the inverse square law.

Applications of Gauss’s Law

1. Field due to an Infinitely Long Straight Uniformly Charged Wire:$E = \frac{\lambda}{2\pi\epsilon_0 r}$
   • Direction is radial.
2. Field due to a Uniformly Charged Infinite Plane Sheet:$E = \frac{\sigma}{2\epsilon_0}$
   • Field is independent of distance from the sheet.
3. Field due to a Uniformly Charged Thin Spherical Shell (radius R):
   • Outside (r > R): $E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$ (acts like a point charge at center).
   • Inside (r < R): $E = 0$ (no charge enclosed).

Questions & Discussion

• Q: Why don't protons fly away from the nucleus despite repulsion?
  • A: They are held by the strong nuclear force, which is effective at sub-atomic ranges ($10^{-14}\,m$).
• Q: Why can one ignore quantisation at macroscopic levels?
  • A: Because the charge involved ($\mu C$) is roughly $10^{13}$ times the electronic charge $e$, making the steps between charge values practically continuous.
• Q: How does a charged comb attract neutral paper?
  • A: The comb's non-uniform electric field polarizes the paper, inducing a dipole moment. The gradient of the field then exerts a net attractive force on this induced dipole.