Chapter 1: Electric Charges and Fields Study Notes

Introduction to Electrostatics

• Everyday Phenomena: Common experiences such as seeing a spark or hearing a crackle when removing synthetic clothes or sweaters, especially in dry weather, are manifestations of electric discharge. Similar effects include the sensation of an electric shock when opening a car door or holding a bus's iron bar after sliding across a seat.
• Underlying Cause: These sensations result from the discharge of electric charges through the human body. These charges accumulate due to the rubbing of insulating surfaces, often referred to as the generation of static electricity.
• Definition of Static: In this context, "static" refers to anything that does not move or fluctuate over time.
• Field of Study: Electrostatics is the branch of physics that deals with the study of forces, fields, and potentials arising from static charges.
• Atmospheric Example: Lightning observed during thunderstorms is a large-scale example of electric discharge in the sky.

Historical Context and the Discovery of Electric Charge

• Discovery: The discovery that amber, when rubbed with wool or silk, attracts light objects is credited to Thales of Miletus, Greece, around $600\,BC$.
• Etymology: The term "electricity" is derived from the Greek word elektron, which translates to "amber."
• Material Pairs: Numerous materials exhibit this property. When rubbed, pairs like glass rods and wool, or plastic rods and cat’s fur, can attract light objects such as straw, bits of paper, or pith balls.
• Observations of Repulsion and Attraction:
  • Two glass rods rubbed with wool repel each other.
  • Two strands of wool or silk used to rub the rods also repel each other.
  • A glass rod and the wool used to rub it attract each other.
  • Two plastic rods rubbed with cat’s fur repel each other but attract the fur.
  • A plastic rod attracts a glass rod but repels the silk/wool used for the glass.
• Conclusion of Two Charges: Years of experimentation established that there are exactly two kinds of electric charge. A body that acquires charge through rubbing is said to be electrified.
• Fundamental Rules:
  1. Like charges repel each other.
  2. Unlike charges attract each other.
• Polarity of Charge: This is the specific property that differentiates the two kinds of charges.
• Neutralization: When an electrified glass rod is brought back into contact with the silk it was rubbed against, they no longer attract or repel other objects. This indicates that unlike charges neutralize or nullify each other's effects.
• Nomenclature: The American scientist Benjamin Franklin named the two types of charges "positive" and "negative." By convention, the charge on a glass rod (or cat’s fur) is positive, while the charge on a plastic rod (or silk) is negative.
• Electrical Neutrality: An object with no net charge is termed electrically neutral.

Detection of Charge: The Gold-Leaf Electroscope

• Apparatus: A simple device used to detect charge on a body. It consists of a vertical metal rod housed in a box with two thin gold leaves attached to the bottom.
• Operation: When a charged object touches the metal knob at the top, charge flows down to the leaves. Because both leaves receive the same type of charge, they repel each other and diverge.
• Indication: The degree of leaf divergence serves as a rough indicator of the amount of charge present.

Microscopic Origin of Charge

• Atomic Structure: All matter consists of atoms and molecules. While materials are normally neutral, they contain balanced positive (protons) and negative (electrons) charges.
• Intermolecular Forces: Forces holding molecules and solids together, adhesive forces of glue, and surface tension are all fundamentally electrical, arising from interactions between charged particles.
• Electrification Process: To charge a neutral body, one kind of charge must be added or removed. In solids, only electrons are transferred because they are less tightly bound than the nuclear protons.
• Transfer Mechanism:
  • Positive Charge: Result of a deficit of electrons.
  • Negative Charge: Result of an excess of electrons.
• Conservation in Rubbing: When a glass rod is rubbed with silk, electrons move from the rod to the silk. No new charge is created; the rod becomes positive, and the silk becomes negative by the same magnitude.

Conductors, Insulators, and Semiconductors

• Conductors: Substances that readily allow the passage of electricity. They contain "free" mobile electrons. Examples include metals, human and animal bodies, and the earth.
• Insulators: Materials that offer high resistance to electricity and do not allow it to pass easily. Examples include glass, porcelain, plastic, nylon, and wood.
• Behavior of Charge:
  • On a conductor, transferred charge distributes itself over the entire surface.
  • On an insulator, transferred charge stays localized at the point of contact.
• Grounding/Earthing: When a conductor is touched, charges may leak through the human body to the ground. This defines why a metal spoon held by hand cannot be easily charged by rubbing unless it has an insulating handle.
• Semiconductors: A third category of materials with electrical resistance intermediate between conductors and insulators.

Basic Properties of Electric Charge

• Point Charges: If the size of charged bodies is negligible compared to the distance between them, total charge is assumed to be concentrated at a single point in space.
• Additivity of Charges: Total charge in a system is the algebraic sum of individual point charges. For a system with charges $q_1, q_2, ..., q_n$, the total charge is:     $Q = q_1 + q_2 + q_3 + ... + q_n$     Proper signs ($+\text{ or } -$) must be used. Unlike mass, which is always positive, charge can be negative.
• Conservation of Charge: Within an isolated system, the total charge remains constant. Charge can be redistributed between bodies, but not created or destroyed. In some natural processes, charged particles are created (e.g., a neutron turning into a proton and electron), but the net charge remains zero.
• Quantisation of Charge: All free charges are integral multiples of a basic unit of charge denoted by $e$. The formula is:     $q = ne$     where $n$ is an integer ($0, \pm 1, \pm 2, ...$).
  • $e = 1.602192 \times 10^{-19}\,C$.
  • Charge on an electron is $-e$; on a proton is $+e$.
  • Quantisation was first suggested by Faraday and demonstrated by Millikan in $1912$.
• Macroscopic vs. Microscopic: At the macroscopic level, charges are large compared to $e$ (e.g., $1\,\mu C$ contains $\approx 10^{13}$ electrons). At this scale, the "grainy" nature of charge is invisible, and charge appears continuous. Quantisation is only critical at the microscopic level involving tens or hundreds of $e$.

Numerical Examples of Charge

• Example 1.1: Time to accumulate 1 Coulomb: If $10^9$ electrons move out of a body per second, find the time to accumulate $1\,C$ on another body.
  • Charge given out per second: $10^9 \times 1.6 \times 10^{-19}\,C = 1.6 \times 10^{-10}\,C/s$.
  • Time required: $\frac{1\,C}{1.6 \times 10^{-10}\,C/s} = 6.25 \times 10^9\,s$.
  • Converted to years: $\frac{6.25 \times 10^9}{365 \times 24 \times 3600} \approx 198\,\text{years}$. Thus, $1\,C$ is a massive unit.
• Example 1.2: Charge in a cup of water: Assuming a cup contains $250\,g$ of water ($H_2O$).
  • Molecular mass of $H_2O = 18\,g$.
  • Molecules in cup: $\frac{250}{18} \times 6.02 \times 10^{23}$.
  • Each molecule has $10$ protons and $10$ electrons.
  • Total charge magnitude: $\frac{250}{18} \times 6.02 \times 10^{23} \times 10 \times 1.6 \times 10^{-19}\,C \approx 1.34 \times 10^7\,C$.

Coulomb’s Law

• Definition: A quantitative statement describing the electrostatic force between two point charges at rest in a vacuum.
• Formula: The magnitude of force ($F$) between two point charges $q_1$ and $q_2$ separated by distance $r$ is:     $F = k \frac{|q_1 q_2|}{r^2}$
• Constants:
  • $k = \frac{1}{4\pi\epsilon_0} \approx 9 \times 10^9\,Nm^2 C^{-2}$.
  • $\epsilon_0$ (permittivity of free space) $= 8.854 \times 10^{-12}\,C^2 N^{-1} m^{-2}$.
• Properties: The force acts along the line joining the two charges. It is attractive for unlike charges and repulsive for like charges.
• Definition of 1 Coulomb: $1\,C$ is the charge that, when placed $1\,m$ from an identical charge in vacuum, experiences a repulsion of $9 \times 10^9\,N$.
• Vector Form:     $\mathbf{F}_{21} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{21}^2} \mathbf{\hat{r}}_{21}$     where $\mathbf{F}_{21}$ is the force on $q_2$ due to $q_1$, and $\mathbf{\hat{r}}_{21}$ is the unit vector from $q_1$ to $q_2$.
• Newton’s Third Law: Electrostatic forces obey Newton's third law: $\mathbf{F}_{12} = -\mathbf{F}_{21}$.
• Comparison to Gravity: Both laws follow the inverse-square dependence. However, electrical force is significantly stronger.
  • Ratio of electric force to gravitational force for an electron and proton: $\approx 2.4 \times 10^{39}$.
  • Ratio for two protons: $\approx 1.3 \times 10^{36}$.

Forces Between Multiple Charges: The Superposition Principle

• Concept: To find the force on a single charge surrounded by multiple other charges, Coulomb's law alone is insufficient.
• Principle: The total force on any one charge is the vector sum of all the forces exerted by other charges, taken individually. The interaction between two charges is unaffected by the presence of a third charge.
• Formula: Total force $\mathbf{F}_1$ on charge $q_1$ due to $n$ charges:     $\mathbf{F}_1 = \mathbf{F}_{12} + \mathbf{F}_{13} + ... + \mathbf{F}_{1n}$$\mathbf{F}_1 = \frac{q_1}{4\pi\epsilon_0} \sum_{i=2}^n \frac{q_i}{r_{1i}^2} \mathbf{\hat{r}}_{1i}$

The Electric Field

• Concept Introduction: A source charge $Q$ creates an "electric field" in the space around it. When a test charge $q$ is placed in this field, it experiences a force.
• Definition: The electric field $\mathbf{E}$ at a point $\mathbf{r}$ is given by:     $\mathbf{E}(\mathbf{r}) = \frac{\mathbf{F}}{q} = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \mathbf{\hat{r}}$
• SI Unit: Newtons per Coulomb ($N/C$) or Volts per meter ($V/m$).
• Independence: The electric field $\mathbf{E}$ depends only on the source charge $Q$ and position $\mathbf{r}$, not on the magnitude of the test charge $q$.
• Symmetry: For a point charge, the field magnitude is constant on the surface of a sphere centered on the charge, exhibiting spherical symmetry.
• Directionality:
  • Positive charge: Field lines point radially outward.
  • Negative charge: Field lines point radially inward.
• Physical Significance: While convenient in electrostatics, the field concept is vital in electromagnetics. It explains the time delay between the motion of one charge and the resulting force on another, occurring at the speed of light ($c$).

Electric Field Lines

• Visualization: Introduced by Faraday as a non-mathematical way to visualize electric fields.
• Definition: A curve drawn such that the tangent at any point indicates the direction of the electric field vector at that point.
• Properties:
  1. Lines start from positive charges and end on negative charges (or extend to infinity).
  2. In charge-free regions, they are continuous curves.
  3. Two lines never cross (the field direction at an intersection would not be unique).
  4. They do not form closed loops (due to the conservative nature of the field).
• Field Strength: Proportional to the density of the lines. Where lines are crowded, the field is strong; where they are spread out, the field is weak.

Electric Flux

• Definition: Analogy to fluid flow. It represents the "flow" of the electric field through an area element.
• Area Vector: An area element $\Delta S$ is treated as a vector $\mathbf{\Delta S} = \Delta S \mathbf{\hat{n}}$, where $\mathbf{\hat{n}}$ is the outward normal.
• Equation: Electric flux $\Delta \Phi$ through area $\Delta S$ is:     $\Delta \Phi = \mathbf{E} \cdot \mathbf{\Delta S} = E \Delta S \cos(\theta)$     where $\theta$ is the angle between $\mathbf{E}$ and the normal to the area.
• Total Flux: $\Phi \approx \sum \mathbf{E} \cdot \mathbf{\Delta S}$. In the limit $\Delta S \rightarrow 0$, this becomes a surface integral.
• SI Unit: $N C^{-1} m^2$.

The Electric Dipole

• Definition: A pair of equal and opposite charges ($q$ and $-q$) separated by a distance $2a$.
• Dipole Moment ($\mathbf{p}$): A vector with magnitude $p = q \times 2a$, directed from $-q$ to $+q$.
• Field on the Axis: For points at distance $r$ from the center on the dipole axis ($r \gg a$):     $\mathbf{E} = \frac{2\mathbf{p}}{4\pi\epsilon_0 r^3}$
• Field on the Equatorial Plane: For points at distance $r$ from the center on the perpendicular bisector ($r \gg a$):     $\mathbf{E} = \frac{-\mathbf{p}}{4\pi\epsilon_0 r^3}$
• Fall-off: The field of a dipole decreases as $1/r^3$, faster than the $1/r^2$ of a point charge.
• Polar Molecules: Molecules like $H_2O$ have permanent dipole moments because the centers of positive and negative charges do not coincide. Non-polar molecules like $CO_2$ only develop dipoles in external fields.

Dipole in a Uniform External Field

• Force: In a uniform field $\mathbf{E}$, the net force on a dipole is zero because $\mathbf{F}_+ = q\mathbf{E}$ and $\mathbf{F}_- = -q\mathbf{E}$.
• Torque ($\tau$): Although the net force is zero, the forces act at different points, creating a torque:     $\mathbf{\tau} = \mathbf{p} \times \mathbf{E}$$\tau = pE \sin(\theta)$     This torque tends to align the dipole with the direction of the field.
• Non-uniform Field: If the field is non-uniform, the dipole experiences both a net torque and a net force.

Continuous Charge Distributions

• Macroscopic Averaging: Treating charge as continuous despite its discrete nature at the microscopic level.
• Types:
  1. Linear Charge Density (\lambda): Charge per unit length ($C/m$).        $\lambda = \frac{\Delta Q}{\Delta l}$
  2. Surface Charge Density (\sigma): Charge per unit area ($C/m^2$).        $\sigma = \frac{\Delta Q}{\Delta S}$
  3. Volume Charge Density (\rho): Charge per unit volume ($C/m^3$).        $\rho = \frac{\Delta Q}{\Delta V}$
• Calculation of Field: The total field is the result of summing (integrating) the contributions of all small charge elements ($dq$) in the distribution.

Gauss’s Law

• Statement: The total electric flux through any closed surface $S$ is equal to $\frac{1}{\epsilon_0}$ times the total charge $q$ enclosed by the surface.     $\Phi = \oint \mathbf{E} \cdot d\mathbf{S} = \frac{q_{\text{enclosed}}}{\epsilon_0}$
• Gaussian Surface: Any hypothetical closed surface used to apply Gauss's Law. It should not pass through discrete point charges.
• Key Points:
  • Truth independent of surface shape or size.
  • Flux is zero if the net enclosed charge is zero.
  • Field $\mathbf{E}$ on the left side is due to all charges (inside and outside), but $q$ on the right is only charges inside.
  • Based on the inverse-square law of Coulomb.

Applications of Gauss’s Law

• Infinitely Long Straight Wire: For a wire with linear charge density $\lambda$, the field at distance $r$ is:     $\mathbf{E} = \frac{\lambda}{2\pi\epsilon_0 r} \mathbf{\hat{n}}$     where $\mathbf{\hat{n}}$ is the radial unit vector. The field falls off as $1/r$.
• Infinite Uniformly Charged Plane Sheet: For a sheet with surface charge density $\sigma$, the field is:     $\mathbf{E} = \frac{\sigma}{2\epsilon_0} \mathbf{\hat{n}}$     The field is uniform and independent of the distance from the sheet.
• Uniformly Charged Thin Spherical Shell:
  • Outside the shell ($r \geq R$): Field is the same as if the entire charge $q$ were concentrated at the center:         $\mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \mathbf{\hat{r}}$
  • Inside the shell ($r < R$): The enclosed charge is zero, so the electric field is zero everywhere inside ($E = 0$).

• Coulomb's Law: $F = k \frac{|q<em>1 q</em>2|}{r^2}$
• Electric Field: $\mathbf{E} = \frac{\mathbf{F}}{q} = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \mathbf{\hat{r}}$
• Total Charge in a System: $Q = q<em>1 + q</em>2 + q<em>3 + … + q</em>n$
• Electric Flux: $\Delta \Phi = \mathbf{E} \cdot \mathbf{\Delta S} = E \Delta S \cos(\theta)$
• Dipole Moment: $p = q \times 2a$
• Torque on a Dipole: $\tau = \mathbf{p} \times \mathbf{E} \quad \text{and} \quad \tau = pE \sin(\theta)$
• Gauss's Law: $\Phi = \oint \mathbf{E} \cdot d\mathbf{S} = \frac{q<em>{\text{enclosed}}}{\epsilon</em>0}$
• Electric Field due to Infinitely Long Straight Wire: $\mathbf{E} = \frac{\lambda}{2\pi\epsilon_0 r} \mathbf{\hat{n}}$
• Electric Field due to Infinite Uniformly Charged Plane Sheet: $\mathbf{E} = \frac{\sigma}{2\epsilon_0} \mathbf{\hat{n}}$
• Electric Field Outside a Uniformly Charged Thin Spherical Shell: $\mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \mathbf{\hat{r}}$
• Electric Field Inside a Uniformly Charged Thin Spherical Shell: $\mathbf{E} = 0$
• Conservation of Charge: Charge can be redistributed but not created or destroyed.
• Quantisation of Charge: $q = ne$ where $e = 1.602192 \times 10^{-19}\,C$.
• Charge Relationship in a Time Example: Time to accumulate 1 Coulomb: $t = \frac{1\,C}{1.6 \times 10^{-10}\,C/s} = 6.25 \times 10^9\,s$