Electricity and Magnetism Fundamentals

Electricity and magnetism, unified as electromagnetism, underpin technologies like motors, transformers, and MRI machines. Electricity governs charge interactions, while magnetism arises from moving charges or magnetic materials. This MathMultiverse guide covers key concepts—Coulomb’s Law, electric fields, potentials, Ohm’s Law, capacitance, magnetic fields, forces, and induction—with detailed examples, formulas, interactive visualizations, and applications.

Coulomb’s Law & Electric Fields

Coulomb’s Law quantifies the force between two point charges:

\[ F = k \frac{|q_1 q_2|}{r^2} \]

Where \( k = 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \), \( q_1, q_2 \) are charges (Coulombs), \( r \) is distance (meters).

Related Formulas:

  • Electric Field: \( E = k \frac{|q|}{r^2} \)
  • Electric Potential: \( V = k \frac{q}{r} \)
  • Ohm’s Law: \( V = I R \)
  • Capacitance: \( C = \frac{Q}{V} \), Energy: \( U = \frac{1}{2} C V^2 \)

Example 1: Coulomb’s Law

Charges \( q_1 = 2 \, \mu\text{C} \), \( q_2 = 3 \, \mu\text{C} \), 1 m apart:

\[ F = (8.99 \times 10^9) \frac{(2 \times 10^{-6})(3 \times 10^{-6})}{1^2} = 0.05394 \, \text{N} \]

Example 2: Electric Field

Field from \( 5 \, \mu\text{C} \) at 2 m:

\[ E = (8.99 \times 10^9) \frac{5 \times 10^{-6}}{2^2} = 11237.5 \, \text{N/C} \]

Example 3: Electric Potential

Potential from \( 4 \, \mu\text{C} \) at 0.5 m:

\[ V = (8.99 \times 10^9) \frac{4 \times 10^{-6}}{0.5} = 71920 \, \text{V} \]

Example 4: Ohm’s Law

Voltage across \( R = 10 \, \Omega \), \( I = 2 \, \text{A} \):

\[ V = 2 \cdot 10 = 20 \, \text{V} \]

Example 5: Capacitance

Capacitor \( C = 2 \, \mu\text{F} \), \( V = 100 \, \text{V} \):

\[ Q = (2 \times 10^{-6}) \cdot 100 = 2 \times 10^{-4} \, \text{C} \] \[ U = \frac{1}{2} (2 \times 10^{-6}) (100)^2 = 0.01 \, \text{J} \]

Electric Field vs. Distance

Electric field from a 5 μC charge vs. distance.

Magnetic Fields & Forces

Magnetic field from a current-carrying wire:

\[ B = \frac{\mu_0 I}{2\pi r} \]

Where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A} \).

Related Formulas:

  • Magnetic Force (Charge): \( F = q v B \sin\theta \)
  • Magnetic Force (Wire): \( F = I L B \sin\theta \)
  • Faraday’s Law: \( \mathcal{E} = - \frac{d\Phi_B}{dt} \), \( \Phi_B = B A \cos\theta \)

Example 1: Magnetic Field

Field at 0.1 m from a wire with \( I = 5 \, \text{A} \):

\[ B = \frac{(4\pi \times 10^{-7})(5)}{2\pi (0.1)} = 1 \times 10^{-5} \, \text{T} \]

Example 2: Magnetic Force on Charge

Force on \( q = 2 \, \mu\text{C} \), \( v = 300 \, \text{m/s} \), \( B = 0.5 \, \text{T} \), \( \theta = 90^\circ \):

\[ F = (2 \times 10^{-6}) \cdot 300 \cdot 0.5 \cdot 1 = 3 \times 10^{-4} \, \text{N} \]

Example 3: Magnetic Force on Wire

Force on 0.2 m wire, \( I = 10 \, \text{A} \), \( B = 0.3 \, \text{T} \), \( \theta = 90^\circ \):

\[ F = 10 \cdot 0.2 \cdot 0.3 \cdot 1 = 0.6 \, \text{N} \]

Example 4: Electromagnetic Induction

Loop (\( A = 0.05 \, \text{m}^2 \)) in field changing from 0.2 T to 0.1 T in 0.01 s (\( \theta = 0^\circ \)):

\[ \Delta\Phi_B = (0.1 - 0.2) \cdot 0.05 \cdot 1 = -0.005 \, \text{Wb} \] \[ \mathcal{E} = - \frac{-0.005}{0.01} = 0.5 \, \text{V} \]

Applications

Electromagnetism drives modern technology.

Electric Motor

Wire (0.1 m, 15 A) in 0.4 T field (\( \theta = 90^\circ \)):

\[ F = 15 \cdot 0.1 \cdot 0.4 \cdot 1 = 0.6 \, \text{N} \]

Transformer

Secondary coil (\( \mathcal{E} = 120 \, \text{V} \), \( R = 60 \, \Omega \)):

\[ I = \frac{120}{60} = 2 \, \text{A} \]

MRI Machine

Field at 0.05 m from wire (\( I = 20 \, \text{A} \)):

\[ B = \frac{(4\pi \times 10^{-7})(20)}{2\pi (0.05)} = 8 \times 10^{-5} \, \text{T} \]

Wireless Charging

EMF from \( \Delta\Phi_B = 0.02 \, \text{Wb} \) in 0.04 s:

\[ \mathcal{E} = - \frac{0.02}{0.04} = 0.5 \, \text{V} \]

Capacitor in Circuit

Capacitor (\( 5 \, \mu\text{F} \), 50 V):

\[ U = \frac{1}{2} (5 \times 10^{-6}) (50)^2 = 0.00625 \, \text{J} \]

Defibrillator

Field between plates (0.02 m, 3000 V):

\[ E = \frac{3000}{0.02} = 150000 \, \text{V/m} \]