1. We have looked at the magnetic field from a single loop of wire
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.

2. We can use Ampere’s law to determine the magnitude of the magnetic field inside an ideal solenoid. (Assume B = 0 outside solenoid) 1 2 3 4
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.

3. 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.
FB – Magnetic flux [Wb]
B – Magnetic field strength [T]
A – Area magnetic field passes through [m2]
Wb – Weber = Tm2
The magnetic flux is similar to the electric flux that we discussed for electric fields.
Closed 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!
Electric field lines all leave the positive charge. There is a net electric flux out of the surface.
Gauss’s Law of Magnetism

4. 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 through
the sphere is
1. moI.
2. moI /(4pR2).
3. 4pR2moI.
4. zero.
5. need more information
Answer: 4. Because magnetic field lines close on themselves, the total
magnetic flux through any closed surface is zero.

5. Ampere’s Law is only valid when the electric field is constant in time
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.
Id – displacement current
The corrected form of Ampere’s Law becomes:
Ampere – Maxwell Law
This is one of the fundamental electromagnetic equations!

6. Types of Magnetic Materials
Magnetic fields are created through the motion of charges
Electron orbiting nucleus
Rotation of electron about its own axis – called “Spin”
Magnetic moments for each of these cases are inversely proportional to mass.
Magnetic moment due to orbit
Magnetic moment due to spin
h – Planck’s Constant
Magnetization – magnetic state of a substance
B – Magnetic flux density or magnetic induction
BM - Magnetic Intensity due to magnetization
H – Magnetic Field Strength
External magnetic field
c – Magnetic Susceptibility – how easy it is to magnetize a substance
mm – Magnetic Permeability – how easily a magnetic field interacts with a substance

7. 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 Ni
Paramagnetic – weak magnetic effects
Used to make permanent magnets
Weak interaction between magnetic moments
An external magnetic field is required to align magnetic domains
c is positive
Diamagnetic – non-magnetic materials
No permanent magnets
Weak anti-alignment when external field is applied – causes repulsion
All substances have some diamagnetic properties, but they are not observable if ferromagnetic or paramagnetic properties exist.