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.