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2. MAGNETOSTATICS

Applied EM by Ulaby, Michielssen and Rava

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Chapter 5 Overview

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Magnetostatics

Biot- Savarts Law

Amperes LawMagnetic flux density, Magnetic

force,

torque and moment, Magnetic dipole, inductor

and inductance

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Electric vs Magnetic

Comparison

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Electric & Magnetic Forces

Electromagnetic

(Lorentz) force

Magnetic force

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Magnetic Force on a Current

Element

Differential force dFm on a differential current I dl:

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Torque

d = moment arm

F = force

T = torque

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agne c orque on urrenLoop

No forces on arms 2 and 4 (

because I and B are parallel, or

anti-parallel)

Magnetic torque:

Area of Loop

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Inclined Loop

For a loop with Nturns and whose surface

normal is at angle theta relative to B

direction:

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Biot-Savart Law

Magnetic field induced by

a differential current:

For the entirelength:

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Magnetic Field due to Current

Densities

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E l 5 2 M ti

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Example 5-2: MagneticField of Linear Conductor

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Magnetic Field of Long

Conductor

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Example 5-3: Magnetic Field of a

Loop

Cont.

dHis in the rz plane , and therefore it hascomponents dHr and dHz

z-components of the magnetic fields due to dl

and dl addbecause they are in the same

direction,but their r-components cancel

Hence for element dl:

Magnitude of field due to dl is

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Example 5-3:Magnetic Field of a Loop(cont.)

For the entire loop:

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Magnetic Dipole

Because a circular loop exhibits a magnetic field pattern similar

to the electric field of an electric dipole, it is called a magnetic

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Forces on Parallel Conductors

Parallel wires attract if their currents are in the same

direction, and repel if currents are in oppositedirections

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Tech Brief 10: Electromagnets

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Magnetic Levitation

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Ampres Law

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InternalMagnetic Field of Long

Conductor

For r< a

Cont.

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ExternalMagnetic Field of Long

Conductor

For r> a

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Magnetic Field of Toroid

Applying Amperes law over contour C:

The magnetic field outside the

toroid is zero. Why?

Amperes law states that the line

integral of H around a closed contour Cis equal to the current traversing the

surface bounded by the contour.

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Magnetic Vector Potential A

Electrostati

csMagnetostati

cs

M ti P ti f

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Magnetic Properties of

Materials

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Magnetic Hysteresis

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Boundary Conditions

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Solenoid

Inside the solenoid:

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Inductance

Magnetic Flux

Flux Linkage

Inductance

Solenoid

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The magnetic field in the region S

between the two conductors is

approximately

Example 5-7: Inductance of Coaxial Cable

Total magnetic flux through S:

Inductance per unit length:

T h B i f 11 I d ti

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Tech Brief 11: Inductive

Sensors

LVDT can measure displacement with submillimeter precision

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Proximity Sensor

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Magnetic Energy Density

Magnetic field in the insulating material is

The magnetic energy stored in the

coaxial cable is

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Summary