Fundamental Laws of Electricity and Magnetism

May 9, 2025

Biot-Savart, Ampere, and Gauss

Introduction to Key Scientists

Hans Christian Oersted (1777-1851)

  • Discovered the relationship between electricity and magnetism.
  • Led to the development of electromagnetism.

Jean Baptiste Biot (1774-1862) and Felix Savart

  • Developed the Biot-Savart Law.
  • Worked at the Polytechnic School and College of France.

Andre Marie Ampere (1775-1836)

  • Developed Ampere's Law.
  • Known for contributions to electromagnetism.

Gauss

  • Developed Gauss's Law in electromagnetism.

Biot-Savart Law

  • Describes the magnetic field generated by an electric current.
  • Formula: ( B = \frac{\mu_0 I}{4\pi} \int \frac{d\mathbf{l} \times \mathbf{\hat{r}}}{r^2} )
  • States that the magnetic field ( B ) at a point in space is a result of the current ( I ) flowing through a wire.
  • The direction of ( B ) is perpendicular to the current and the position vector.

Ampere’s Law

  • Relates the magnetic field in terms of an electric current passing through a closed loop.
  • ( \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I )
  • Used to calculate magnetic fields in symmetric geometries.

Applications

  • Solenoids and toroids.
  • Magnetic field calculations for closed loops.

Gauss’s Law

  • Describes the distribution of electric charge to the resulting electric field.
  • States that the total electric flux out of a closed surface is equal to the charge enclosed divided by the electric constant.

Formula

  • ( \Phi = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\epsilon_0} )_

Laplace's Force

  • Acts on a current-carrying wire in a magnetic field.
  • Force formula: ( F = BIl \sin \theta )

Important Concepts

Magnetic Field

  • Measured in Tesla (T).
  • Direction given by right-hand rule.

Applications

  • Electromagnetic devices such as motors and generators.
  • Concepts underpinning magnetic materials and forces.

Hall Effect

  • Discovered by Edwin Hall in 1879.
  • Demonstrates the force experienced by charge carriers in a conductor in a magnetic field.
  • Hall voltage is used to measure magnetic field strength.

Formula

  • ( V_H = \frac{IB}{nqAd} )

Key Takeaways

  • Understanding of the interactions between electricity and magnetism expanded through work by Oersted, Biot-Savart, Ampere, and Gauss.
  • These principles form the foundation for modern electrical and magnetic applications.