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.