Lecture 9 Vector Magnetic Potential Biot Savart Law Prof. Viviana Vladutescu

Figure 1: The magnetic (H-field) streamlines inside and outside a single thick wire.

Figure 2: The H-field magnitude inside and outside the thick wire with uniform current density

Figure 3: The H-field magnitude inside and outside the thick conductors of a coaxial line.

Vector Magnetic Potential A - vector magnetic potential (Wb/m)

Figure 1: The vector potential in the cross-section of a wire with uniform current distribution.

Figure 2: Comparison between the magnetic vector potential component  of a wire with uniformly distributed current and the electric potential V of the equivalent cylinder with uniformly distributed charge.

Poisson’s Equation Laplacian Operator (Divergence of a gradient) Vector Poisson’s equation

In electrostatics Poisson’s Equation in electrostatics

Magnetic Flux The line integral of the vector magnetic potential A around any closed path equals the total magnetic flux passing through area enclosed by the path

Biot Savart Law and Applications

The Biot-Savart Law relates magnetic fields to the currents which are their sources. In a similar manner, Coulomb’s Law relates electric fields to the point charges which are their sources. Finding the magnetic field resulting from a current distribution involves the vector product, and is inherently a calculus problem when the distance from the current to the field point is continuously changing.

Biot-Savart Law By using (see eq 6.31)

In two steps

Illustration of the law of Biot–Savart showing magnetic field arising from a differential segment of current.

Example1 Component values for the equation to find the magnetic field intensity resulting from an infinite length line of current on the z-axis. (ex 6-4)

Example 2 We want to find H at height h above a ring of current centered in the x – y plane.

The component values shown for use in the Biot–Savart equation.

The radial components of H cancel by symmetry.

Solenoid Many turns of insulated wire coiled in the shape of a cylinder.

For a set N number of loops around a ferrite core, the flux generated is the same even when the loops are bunched together.

Example : A simple toroid wrapped with N turns modeled by a magnetic circuit. Determine B inside the closely wound toroidal coil. b a

Ampere’s Law

Electromagnets a) An iron bar attached to an electromagnet. b) The bar displaced by a differential length d.

Applications Levitated trains: Maglev prototype Electromagnet supporting a bar of mass m.

Wilhelm Weber (1804-1891). Electromagnetism.