Electrical Energy Technology

Electrical Energy Technology

Electrical Energy Technology

Electric Charge, Force and Field Guide

Contents

1. Introduction

The electromagnetic force is one of the four fundamental forces in the universe. The three other fundamental forces are:

  • The gravitational force
  • The strong force between nuclear particles
  • The weak force between nuclear particles

The electromagnetic force is important in chemistry. It is the glue that holds atoms together in molecules and crystals including metal crystals.

Almost all devices in the modern world are based on charged particles in motion. Charged particles are influenced by the electromagnetic force, which alters their motion. An analogy is a roller coaster in an amusement park - just as mass is accelerated by gravitational force, a charged particle is accelerated by electromagnetic force.

Practical Example: Picture Tube

In a picture tube, electrons that emerge from the heated cathode are:

  • First accelerated by an electrostatic force
  • Then deflected by a magnetic force

Fundamental Constants:

  • Vacuum permittivity: ε₀ = 8.85 × 10⁻¹² C²/Nm²
  • Vacuum permeability: μ₀ = 4π × 10⁻⁷ Ns²/C²
  • Relationship with speed of light: c² = 1/(ε₀μ₀)

6. The E-field from a Line of Charge

Consider a rod with a total charge Q uniformly distributed along its length L=2a. The linear charge density is λ = Q/L.

Key Equations:

\[ dQ = \lambda dy = Q dy / L \] \[ dE = \frac{dQ}{4\pi\epsilon_0(x^2 + y^2)} = \frac{Q dy}{8a\pi\epsilon_0(x^2 + y^2)} \] Final result for field at point P: \[ E_x = \frac{Q}{4\pi\epsilon_0x(x^2 + a^2)^{1/2}} \]

Special Case:

For points close to a very long line charge (x≪a):

\[ E_x = \frac{\lambda}{2\pi\epsilon_0x} \]

Note: The field depends on 1/x rather than 1/x²

7. The E-field from a Ring of Charge

A ring-shaped conductor carries a total charge dQ. We calculate the electric field at a point P on the axis of the ring.

Field Equations:

\[ E_x = \frac{x dQ}{4\pi\epsilon_0(x^2 + r^2)^{3/2}} \] \[ E_y = 0 \]

Limiting Cases:

  • At x = 0: E_x = 0
  • As x → ∞: Ring acts like a point charge

8. The E-field from a Disc of Charge

We consider a disc of radius R with uniform surface charge density σ = Q/πR².

Field Expression:

\[ E_x = \frac{\sigma}{2\epsilon_0}\left\{1 - \frac{1}{\left(1 + \frac{R^2}{x^2}\right)^{1/2}}\right\} \]

Important Result:

For points close to a large plane surface (x → 0):

\[ E_x = \frac{\sigma}{2\epsilon_0} \]

9. The Electric Dipole

An electric dipole consists of two equal but opposite charges (+q and -q) separated by a distance d.

Key Concepts:

  • Electric dipole moment: \(\vec{p_e} = q\vec{d}\)
  • Torque: \(\vec{\tau} = \vec{p_e} \times \vec{E}\)

Important Properties:

  • Stable when aligned with field (θ = 0°)
  • Maximum torque when perpendicular to field (θ = 90°)
  • Unstable when anti-aligned (θ = 180°)

Real-World Example:

Water molecules are electric dipoles. The oxygen atom gains electrons from the hydrogen atoms, creating a negative charge at O and positive charges at H atoms.

10. Problems & Test Questions

Practice Problems

Problem 2.1: Vector Addition of Electric Forces Along a Line

Given charges: q₀ = 1 pC, q₁ = 1 nC, q₂ = -1 nC
Distances: r₁ = 1 cm, r₂ = 2 cm
Calculate the total vector force on q₀

Problem 2.2: Vector Addition of Electric Forces in a Plane

Three charges in xy-plane:
q₀ = 1 pC, q₁ = 1 nC, q₂ = -1 nC
Calculate total force on q₀

Test Questions (Sample)

  • Explain Coulomb's law (20p)
  • Define the electric field (10p)
  • Express electric force using E-field and charge (10p)
  • Derive E-field from a point charge (10p)
  • Explain E-field line patterns (10p)

2. Basic Electrostatic Phenomenon

Charge (electrons) can be transferred from one object to another when we rub:

  • Plastic rods against fur
  • Glass rods against silk

Key Points:

  • Charged objects with the same algebraic sign repel
  • Charged objects with opposite algebraic sign attract
  • The charge quantum is e = 1,602 x 10⁻¹⁹ C

Atomic Structure and Charge

An atom consists of:

  • A nucleus containing positively charged protons and uncharged neutrons
  • Negatively charged electrons orbiting the nucleus
  • In a neutral atom, the number of electrons equals the number of protons

Ionization:

  • Gaining electrons → Negative ion
  • Losing electrons → Positive ion
  • Size of atom (with electrons): ~0.1 nm
  • Size of nucleus: ~1 fm

Conductors and Insulators

Metals (Conductors):

  • Form crystal lattices
  • One or more outer electrons per atom become free to move
  • These "conduction electrons" can move throughout the crystal
  • Number of free electrons equals number of atoms

Insulators:

  • All electrons are bound to specific atoms
  • Electrons can only move short distances within atoms or molecules
  • No free electrons for conduction

Conservation of Charge:

In a closed system, charge cannot be created or destroyed. When objects become charged, it's through transfer of existing charges:

  • Electrons transferring from fur to plastic rod
  • Total positive charge equals total negative charge transferred
  • Charge is always quantized in multiples of e

3. How to Charge an Object

Charging by Contact

When a negatively charged plastic rod is connected to a metal sphere via a copper wire, negative charge can be transferred to the metal sphere. After removing the copper wire, the negatively charged rod will repel the negatively charged metal sphere.

Key Properties:

  • Conductors permit easy motion of charge
  • Insulators do not permit easy motion of charge
  • Metals are typically good conductors
  • Non-metals are typically good insulators

Charging by Induction

In this method, an object can be charged without direct contact. When a charged rod is brought near a metal sphere, it causes a redistribution of charges within the sphere. The sphere can then be permanently charged by grounding it while under the influence of the charged rod.

Polarization

When a charged object (like a comb) is brought near neutral pieces of paper or plastic, it can cause a slight shifting of charge within the molecules of the neutral material. This effect, called polarization, results in an attractive force even though the overall charge of the material remains neutral.

4. Coulomb's Law

Coulomb (1736-1806) derived an expression for the force between charged particles. For two charges q and q₀, he found that the force is proportional to both charges and inversely proportional to the squared distance between them.

The Mathematical Expression:

\[ F_e = k_e \frac{|q \cdot q_0|}{r^2} \] where:
  • F_e is the electric force
  • k_e is Coulomb's constant (8.99 × 10⁹ Nm²/C²)
  • q and q₀ are the charges
  • r is the distance between charges

Important Properties:

  • The force is repulsive for like charges
  • The force is attractive for opposite charges
  • The force follows the inverse square law

Historical Measurement: The Torsion Balance

Coulomb used a torsion balance (1777) to measure electric forces. The apparatus consisted of:

  • Two spheres connected by an insulating beam
  • The beam suspended by a thin filament
  • A third sphere fixed to a conducting rod

Working Principle:

  • Torque: τ = κθ
  • κ is the torsion constant of the filament
  • θ is the twist angle
  • Force can be calculated from: F = κθ/R

Vector Form and Superposition

Vector Form:

\[ \vec{F_e} = k_e \frac{q \cdot q_0}{r^2} \hat{r} \]

where \(\hat{r}\) is the unit vector from q to q₀

Principle of Superposition:

For multiple charges, the total force is the vector sum:

\[ \vec{F_e} = \vec{F_1} + \vec{F_2} + ... + \vec{F_N} \]

5. The E-field

Definition of Electric Field

The electric field at a point P is defined as the electric force F_e on a test charge q₀ divided by q₀:

\[ \vec{E} \equiv \frac{\vec{F_e}}{q_0} \]

The unit is N/C (Newtons per Coulomb)

Electric Field Lines

Electric field lines are imaginary curves that help visualize the electric field. Key properties include:

  • Lines point away from positive charges
  • Lines point toward negative charges
  • The field lines never intersect
  • The density of field lines indicates field strength

Special Cases:

  • Point charge: field lines radiate outward uniformly
  • Dipole: field lines start at positive charge, end at negative charge
  • Uniform field: parallel, equally spaced field lines

Electric Field from a Point Charge

\[ \vec{E} = k_e \frac{q}{r^2} \hat{r} \]

where \(\hat{r}\) is the unit vector pointing from the charge to the field point

Electrical Energy Technology

Electrical Energy Technology