Electric Charges and Fields Study Notes

Introduction to Electrostatics

• Phenomenological Observations: Humans experience sparks or crackling sounds when removing synthetic clothes (like sweaters) in dry weather. This is attributed to electric discharge of charges accumulated due to the rubbing of insulating surfaces.

• Lightning: A large-scale common example of electric discharge occurring during thunderstorms.

• Electric Shock: Sensations of shock when opening car doors or holding iron bars on buses after sliding across seats are due to the discharge of accumulated static electricity through the body.

• Static Electricity: "Static" refers to anything that does not move or change with time.

• Definition of Electrostatics: This branch of physics deals with the study of forces, fields, and potentials arising from static (not moving) charges.

Electric Charge and Historical Discovery

• Thales of Miletus: A Greek philosopher credited with the discovery (around 600 BC) that amber rubbed with wool or silk cloth attracts light objects.

• Etymology: The word "electricity" is derived from the Greek word elektron, which means amber.

• Experimental Observations:

  • Two glass rods rubbed with wool or silk repel each other.

  • The two strands of wool or silk used for rubbing 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 on the glass.

• Fundamental Conclusions: There are only two kinds of electric charge. Like charges repel, and unlike charges attract.

• Polarity of Charge: The property that differentiates the two kinds of charges.

• Neutralization: When electrified bodies are brought into contact, they lose their charge and no longer attract/repel objects. This implies that unlike charges nullify each other.

• Naming Convention: Benjamin Franklin named the charges "positive" and "negative."

  • By convention, the charge on a glass rod (or cat’s fur) is positive.

  • The charge on a plastic rod (or silk) is negative.

• Neutral State: An object with no net charge is termed electrically neutral.

Detection of Charge: The Gold-Leaf Electroscope

• Apparatus Description: A vertical metal rod housed in a box with two thin gold leaves attached at the bottom.

• Mechanism: When a charged object touches the metal knob at the top, charge flows to the leaves, causing them to diverge due to repulsion.

• Indication: The degree of divergence indicates the amount of charge present.

The Origin of Charge in Solids

• Atomic Structure: Matter consists of atoms and molecules. Normally, the positive charges (protons) and negative charges (electrons) are exactly balanced.

• Charging Mechanism: To electrify a neutral body, one must add or remove charges. In solids, electrons are less tightly bound and are the particles transferred between bodies during rubbing.

• Charge Types by Deficit/Excess:

  • Positively Charged: A body that has lost some of its electrons.

  • Negatively Charged: A body that has gained electrons.

• Conservation during Rubbing: No new charge is created. For example, when a glass rod is rubbed with silk, electrons transfer from the rod to the silk; the rod becomes positive and the silk becomes negative by the exact same amount.

Conductors, Insulators, and Semiconductors

• Conductors: Substances that allow electricity to pass through them easily. They have free-moving electrons.

  • Examples: Metals, human and animal bodies, and the earth.

• Insulators: Substances that offer high resistance to the passage of electricity. Charges placed on them stay at the same location.

  • Examples: Glass, porcelain, plastic, nylon, and wood.

• Semiconductors: A third category offering resistance intermediate between conductors and insulators.

• Charge Distribution: When charge is transferred to a conductor, it distributes over the entire surface. On an insulator, it remains localized.

• Earthing/Grounding: The process of sharing charges with the earth. When a metal spoon is held by hand while rubbing, the charge leaks through the body to the ground. However, a metal rod with an insulating (wooden/plastic) handle can be charged.

Basic Properties of Electric Charge

• Point Charges: If the size of charged bodies is much smaller than the distance between them, they are treated as point charges (concentrated at a single point in space).

• Additivity of Charges: Total charge in a system is the algebraic sum of all individual charges.

  • If a system has charges $q_1, q_2, \dots, q_n$, then total charge $Q = q_1 + q_2 + \dots + q_n$.

  • Example: A system with charges $+1, +2, -3, +4, -5$ has a total charge of $(+1) + (+2) + (-3) + (+4) + (-5) = -1$.

• Conservation of Charge: The total charge of an isolated system remains constant. Charged particles may be created or destroyed (e.g., a neutron turning into a proton and an electron), but the net charge remains zero if the system was neutral.

• Quantisation of Charge: All free charges are integral multiples of a basic unit of charge denoted by $e$.

  • Formula: $q = ne$, where $n$ is an integer ($n = \pm 1, \pm 2, \dots$).

  • Basic Unit ($e$): The charge of an electron ($-e$) or proton ($+e$).

  • SI Unit: The Coulomb ($C$).

  • Value of $e$: $e = 1.602192 \times 10^{-19}\,C$.

  • Scale of $-1\,C$: Contains approximately $6 \times 10^{18}$ electrons.

  • Microscopic vs. Macroscopic: At the macroscopic level (where charges are $\mu C$), the grainy nature of charge is ignored, and distribution appears continuous. At the microscopic level (tens or hundreds of $e$), quantisation is essential.

Coulomb’s Law

• Definition: The electrostatic force between two point charges is inversely proportional to the square of the distance between them and directly proportional to the product of their magnitudes. It acts along the line joining the charges.

• Magnitude Equation:     $F = k \frac{|q_1 q_2|}{r^2}$

• The Constant $k$: In vacuum, $k = \frac{1}{4\pi\epsilon_0}$.

  • Value: $k \approx 9 \times 10^9\,Nm^2/C^2$.

• Permittivity of Free Space ($\epsilon_0$):     $\epsilon_0 = 8.854 \times 10^{-12}\,C^2 N^{-1}m^{-2}$

• Vector Form:     $\mathbf{F}_{21} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{21}^2} \mathbf{\hat{r}}_{21}$

  • $\mathbf{F}_{21}$ is the force on charge $q_2$ due to charge $q_1$.

  • $\mathbf{\hat{r}}_{21}$ is the unit vector from 1 to 2.

  • $\mathbf{F}_{21} = -\mathbf{F}_{12}$ (Consistent with Newton’s Third Law).

Comparison of Forces: Electrostatic vs. Gravitational

• Similarity: Both follow the inverse-square law ($F \propto 1/r^2$).

• Difference: Gravity is always attractive; electrostatic forces can be attractive or repulsive.

• Strength Comparison (Example 1.3):

  • For an electron and a proton, $\frac{F_e}{F_G} \approx 2.4 \times 10^{39}$.

  • For two protons, $\frac{F_e}{F_G} \approx 1.3 \times 10^{36}$.

• Particle Acceleration: Due to the high strength of electric forces, particles like electrons experience enormous accelerations (e.g., $2.5 \times 10^{22}\,m/s^2$ in a basic atomic field), making gravity negligible in atomic physics.

Superposition Principle

• Definition: The force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges taken one at a time.

• Equation for charge $q_1$:     $\mathbf{F}_1 = \mathbf{F}_{12} + \mathbf{F}_{13} + \dots + \mathbf{F}_{1n}$     $\mathbf{F}_1 = \frac{1}{4\pi\epsilon_0} \sum_{i=2}^{n} \frac{q_1 q_i}{r_{1i}^2} \mathbf{\hat{r}}_{1i}$

The Electric Field

• Concept: A charge $Q$ produces an electric field ($\mathbf{E}$) everywhere in its surroundings. When a test charge $q$ is placed at a point, the field acts on it to produce a force.

• Mathematical Definition:     $\mathbf{E}(\mathbf{r}) = \frac{\mathbf{F}}{q} = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \mathbf{\hat{r}}$

• Units: Newtons per Coulomb ($N/C$) or Volts per meter ($V/m$).

• Source vs. Test Charge:

  • Source Charge ($Q$): The charge creating the field.

  • Test Charge ($q$): Used to measure the field; should be infinitesimally small ($q \rightarrow 0$) so it does not disturb the source charge position.

• Symmetry: For a point charge, the field has spherical symmetry; the magnitude depends only on distance $r$.

• Physical Significance: In time-dependent situations, electromagnetic fields propagate at the speed of light ($c$). The field accounts for the time delay between the motion of one charge and the force on another. Fields transport energy.

Electric Field Lines

• Definition: A pictorial representation of the electric field. It is a curve where the tangent at any point gives the direction of the net electric field at that point.

• Properties:

  1. Lines start at positive charges and end at negative charges. They can start or end at infinity for single charges.

  2. In charge-free regions, they are continuous curves without breaks.

  3. Two field lines can never cross (intersection would mean two directions for the net field, which is impossible).

  4. They do not form closed loops (electrostatic fields are conservative).

• Density and Strength: Closeness of lines indicates field strength. Higher density of lines per unit cross-sectional area implies a stronger field.

Electric Flux

• Definition: A measure of the total number of field lines crossing a surface.

• Equation for Area Element ($\Delta S$):     $\Delta \phi = \mathbf{E} \cdot \Delta \mathbf{S} = E \Delta S \cos(\theta)$

  • $\theta$ is the angle between $\mathbf{E}$ and the outward normal to the surface.

• Area Vector: By convention, for a closed surface, the area vector points in the direction of the outward normal.

• Units: $N C^{-1} m^2$.

The Electric Dipole

• Definition: A pair of equal and opposite charges ($q$ and $-q$) separated by a distance $2a$.

• Dipole Moment ($\mathbf{p}$):     $\mathbf{p} = q \times 2a\,\mathbf{\hat{p}}$

  • Direction is from $-q$ to $q$.

• Electric Field of a Dipole at Large Distances ($r \gg a$):

  • On the Axis:         $\mathbf{E} = \frac{2\mathbf{p}}{4\pi\epsilon_0 r^3}$

  • On the Equatorial Plane:         $\mathbf{E} = -\frac{\mathbf{p}}{4\pi\epsilon_0 r^3}$

• Point Dipole: The limit where $2a \rightarrow 0$ and $q \rightarrow \infty$ while $p$ remains finite.

• Dipole in Uniform External Field:

  • Force: The net force is zero ($q\mathbf{E} + (-q\mathbf{E}) = 0$).

  • Torque ($\tau$):         $\tau = \mathbf{p} \times \mathbf{E}$         $\tau = p E \sin(\theta)$

  • Torque Effect: Tends to align the dipole with the direction of the field.

  • Non-uniform Field: Dipole experiences both torque and a net force.

Continuous Charge Distributions

• Linear Charge Density ($\lambda$): Charge per unit length ($C/m$).     $\lambda = \frac{\Delta Q}{\Delta l}$

• Surface Charge Density ($\sigma$): Charge per unit area ($C/m^2$).     $\sigma = \frac{\Delta Q}{\Delta S}$

• Volume Charge Density ($\rho$): Charge per unit volume ($C/m^3$).     $\rho = \frac{\Delta Q}{\Delta V}$

• Calculation of Field:     $\mathbf{E} \approx \frac{1}{4\pi\epsilon_0} \sum_{all \, \Delta V} \frac{\rho \, \Delta V}{r^2} \mathbf{\hat{r}}$

Gauss’s Law

• Statement: The total electric flux through any closed surface $S$ is equal to $1/\epsilon_0$ times the total charge enclosed by that surface.

• Equation:     $\phi = \frac{q_{enclosed}}{\epsilon_0}$

• Gaussian Surface: An imaginary closed surface used for calculating the flux. It should not pass through discrete charges but can pass through continuous distributions.

• Properties of the Law:

  1. True for any closed surface of any shape or size.

  2. $q$ is the net charge (sum of all charges) inside.

  3. If net flux is zero, the net charge inside is zero.

  4. Useful for determining fields of symmetric configurations.

  5. Based on the inverse-square law of Coulomb.

Applications of Gauss’s Law

• Infinitely Long Straight Uniformly Charged Wire:     $\mathbf{E} = \frac{\lambda}{2\pi\epsilon_0 r} \mathbf{\hat{n}}$

  • Field is radial and depends on $1/r$.

• Uniformly Charged Infinite Plane Sheet:     $\mathbf{E} = \frac{\sigma}{2\epsilon_0} \mathbf{\hat{n}}$

  • Field is independent of distance from the sheet.

• Uniformly Charged Thin Spherical Shell (Radius $R$):

  • Outside ($r \geq R$):         $\mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \mathbf{\hat{r}}$         (Field is as if the total charge $q$ is concentrated at the center).

  • Inside (r < R):         $\mathbf{E} = 0$         (Experimental verification of this zero field confirms the $1/r^2$ dependence in Coulomb's Law).

Selected Mathematical Examples

• Example 1.1: Time to collect $1\,C$ of charge if $10^9$ electrons transfer per second:

  • Electronic charge transfer rate = $(10^9) \times (1.6 \times 10^{-19}) = 1.6 \times 10^{-10}\,C/s$.

  • Time = $1\,C / (1.6 \times 10^{-10}\,C/s) = 6.25 \times 10^9\,s \approx 198\,years$.

• Example 1.2: Charge in a 250 g cup of water:

  • Moles of water = $250/18$.

  • Molecules = $(250/18) \times 6.02 \times 10^{23}$.

  • Each molecule has 10 protons and 10 electrons.

  • Total positive charge $Q = (250/18) \times 6.02 \times 10^{23} \times 10 \times 1.6 \times 10^{-19} \approx 1.34 \times 10^7\,C$.

• Example 1.10: Flux through a cube ($a = 0.1\,m$) in field $E_x = \alpha x^{1/2}$, $\alpha = 800\,N/C\,m^{1/2}$:

  • $E_L = \alpha a^{1/2}$, $E_R = \alpha (2a)^{1/2}$.

  • $Net \, Flux \, \phi = a^2 (E_R - E_L) = \alpha a^{5/2} (\sqrt{2} - 1) = 1.05\,Nm^2/C$.

  • $Charge \, q = \phi \times \epsilon_0 = 1.05 \times 8.854 \times 10^{-12} = 9.27 \times 10^{-12}\,C$.