Electricity and Magnetism

Electricity and magnetism are interconnected phenomena that govern many aspects of the physical world, from lightning to motors.

Coulomb’s Law

The force between two point charges is proportional to their magnitudes and inversely proportional to the square of the distance between them:

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

Where \(k\) is Coulomb’s constant (\(8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2\)), \(q_1\) and \(q_2\) are charges, and \(r\) is distance.

Example: Two charges, \(2 \, \mu\text{C}\) and \(3 \, \mu\text{C}\), 1 m apart:

\[ F = 8.99 \times 10^9 \frac{(2 \times 10^{-6})(3 \times 10^{-6})}{1^2} = 0.0539 \, \text{N} \]

A current-carrying wire generates a magnetic field, described by the Biot-Savart Law. For a long straight wire:

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

Where \(\mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A}\), \(I\) is current, and \(r\) is distance.

Example: \(I = 5 \, \text{A}\), \(r = 0.1 \, \text{m}\):

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

Electromagnetism powers electric motors, transformers, MRI machines, and wireless communication.