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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 .
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Lecture 9 Vector Magnetic PotentialBiot 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
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)
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.
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.
a b Example : A simple toroid wrapped with N turns modeled by a magnetic circuit. Determine B inside the closely wound toroidal coil.
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.
