We have looked at the magnetic field from a single loop of wire. It is also necessary to examine the magnetic field from a stack of looped coils. This stack is really just a wire wound into a helical shape. This configuration of looped wire is called a solenoid.
The concentration of magnetic field lines is greater in the center of the solenoid, than it is outside the solenoid. This means that there will be a larger net magnetic field inside the solenoid than outside. The magnetic field for a solenoid resembles that of a bar magnet.The magnetic field will be different if the loops are closely spaced as compared to widely separated loops.The solenoid is the most commonly discussed and used loop configuration. A solenoid bent into a circle is called a toroid.
We can use Ampere’s law to determine the magnitude of the magnetic field inside an ideal solenoid. (Assume B = 0 outside solenoid)
x
x
x
x
x
I B
W
1
2
3
4
0 enclB ds I
1 2 3 4 0 enclB ds B ds B ds B ds I
0
2B ds
04B ds
00outsideB
0 enclB I enclI NI
0 enclIB
0NI
0B nI
Nn
Magnetic field at the center of an ideal solenoid
This equation is only valid at the center of a solenoid assuming that there is no magnetic field outside the solenoid and the coils are closely spaced.
A solenoid has a magnetic field that through its center, which means that it passes through an area defined by the geometry of the loops. This means that there is a Magnetic Flux through the solenoid.
Magnetic Flux – The amount of magnetic field that passes through a specified area.
The magnetic flux is similar to the electric flux that we discussed for electric fields.
B B dA FB – Magnetic flux [Wb]
B – Magnetic field strength [T]A – Area magnetic field passes through [m2]
Wb – Weber = Tm2
N
S
Closed Surface
Electric field lines all leave the positive charge. There is a net electric flux out of the surface.
The same number of magnetic field lines enter the closed surface as leave.
The net flux through a closed surface must always be zero!
0B B dA
Gauss’s Law of Magnetism
A sphere of radius R is placed near a long, straight wire that carries a steady current I. The magnetic field generated by the current is B. The total magnetic flux passing throughthe sphere is
1. moI.2. moI /(4pR2).3. 4pR2moI.4. zero.5. need more information
Ampere’s Law is only valid when the electric field is constant in time. Time-varying electric fields are a common occurrence though and must be discussed. When a time-varying electric field is present Ampere’s Law will still be valid if we include a correction term to account for the time-varying electric field.
Constant electric field - electric field from a power supply pushing charges in a single direction.Time-varying electric field – Electric field generated by a charging or discharging capacitor
The time-varying component of the electric field causes a secondary current called the displacement current.
0E
d
dI
dt
Id – displacement current
The corrected form of Ampere’s Law becomes:
0 0 0 0E
d
dB ds I I I
dt
Ampere – Maxwell Law
This is one of the fundamental electromagnetic equations!
Types of Magnetic MaterialsMagnetic fields are created through the motion of charges
• Electron orbiting nucleus• Rotation of electron about its own axis – called “Spin”
2
e
m
Magnetic moments for each of these cases are inversely proportional to mass.
Magnetic moment due to orbit
2spin
e
m
Magnetic moment due to spin
341.05 102
hx Js
h – Planck’s Constant
Magnetization – magnetic state of a substance MV
0 MB B B
0MB M
B – Magnetic flux density or magnetic inductionBM - Magnetic Intensity due to magnetizationH – Magnetic Field Strength
0
0
BH
0 0H M 0 H M
M H
c – Magnetic Susceptibility – how easy it is to magnetize a substance
0 1m mm – Magnetic Permeability – how easily a magnetic field interacts with a substance
External magnetic field
Classes of Magnetic Materials
Ferromagnetic – Crystalline substances with strong magnetic effects
• Used to make permanent magnets• Strong interaction between magnetic moments
Examples: Fe, Co and NiParamagnetic – weak magnetic effects
• Used to make permanent magnets• Weak interaction between magnetic moments• An external magnetic field is required to align magnetic domains
0 1 c is positive
0m
Diamagnetic – non-magnetic materials• No permanent magnets• Weak anti-alignment when external field is applied – causes repulsion
0m
All substances have some diamagnetic properties, but they are not observable if ferromagnetic or paramagnetic properties exist.
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