# Electrical and Computer Engineering Dept.

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Electrical and Computer Engineering Dept.
Magnetism INEL 4151 Dr. Sandra Cruz-Pol Electrical and Computer Engineering Dept. UPRM ch 7

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Applications Motors Transformers MRI More…

H= magnetic field intensity [A/m] B= magnetic field density [Teslas]
In free space the permeability is:

Magnetic Field Biot-Savart Law
States that:

Example For an infinite line filament with current I
af a1 r For an infinite line filament with current I (a1=180o and a2=0o):

PE. 7.1 Find H at (0,0,5) Due to current in (figure): where a1=90o and
z Due to current in (figure): where a1=90o and (0,0,5) y 10A 1 x

Circular loop Defined by Apply Biot-Savart:
dl R z y dHz dHr x Defined by Apply Biot-Savart: Only z-component of H survives due to symmetry:

Ampere’s Law Simpler Analogous to Gauss Law for Coulombs
For symmetrical current distributions

Ampere’s Law We define an Amperian path where H is constant.

Infinitely long coaxial cable
z Four cases: 1) For r<a

Infinitely long coaxial cable
z Four cases: 2) For a<r<b

Infinitely long coaxial cable
Four cases: 3) For b<r<b+c z

Infinitely long coaxial cable
Four cases: 4) For r>b+c

Sheet of current distribution
z Cross section is a Line! 4 b a The H field is given by: 1 y K [A/m] 3 2 The H field on the Amperian path is given by: x

PE. 7.5 Sheet of current Plane y=1 carries a current K=50 az mA/m. Find H at (0,0,0). y K =50 mA/m z -x

A toroid A circular ring-shaped magnetic core of iron powder, ferrite, or other material around which wire [N- loops] is coiled to make an inductor. Toroidal coils are used in a broad range of applications, such as high-frequency coils and transformers. Toroidal inductors can have higher Q factors and higher inductance than similarly constructed solenoid coils. This is due largely to the smaller number of turns required when the core provides a closed magnetic path. The magnetic flux in a toroid is largely confined to the core, preventing its energy from being absorbed by nearby objects, making toroidal cores essentially self-shielding. Fields stay inside core, no interference.

Magnetic Flux Density, B
The magnetic flux is defined as: which flows through a surface S. The total flux thru a closed surface in a magnetic field is: Monopole doesn’t exist.

Maxwell’s Equations for Static Fields
Differential form Integral Form Gauss’s Law for E field. Gauss’s Law for H field. Nonexistence of monopole Faraday’s Law; E field is conserved. Ampere’s Law

Magnetic Scalar and Vector Potentials, Vm & A
When J=0, the curl of H is =0, then recalling the vector identity: We can define a Magnetic Scalar Potential as: The magnetic Vector Potential A is defined:

The magnetic vector potential, A, is
It can be shown that: The magnetic vector potential A is used in antenna theory. Substituting into equation for Magnetic Flux: This is another way of finding magnetic flux.

P.E. 7.7 A current distribution causes a magnetic vector potential of:
Find : B at (-1,2,5) Answer: Flux thru surface z=1, 0≤x≤1, -1≤y ≤4 Answer :