1. Magnetic Sources The Biot-Savart Law
Magnetic fields go around the wire – they are perpendicular to the direction of current
Magnetic fields are perpendicular to the separation between the wire and the point where you measure it
Sounds like a cross product! r I ds
Permeability of free space
The Amp is defined to work out this way

2. Sample Problem
A loop of wire consists of two quarter circles of radii R and 2R, both centered at a point P, and connected with wires going radially from one to the other. If a current I flows in the loop, what is the magnetic field at point P? ds ds I ds
Do one side at a time
First do one of the straight segments
ds and r-hat are parallel
No contribution to the integral
Other straight segment is the same P ds R 2R
Now do inner quarter loop
Outer loop opposite direction, similar

3. Ampere’s Law (original recipe)
Suppose we have a wire coming out of the plane
Let’s integrate the magnetic field around a closed path
There’s a funky new symbol for such an integral
Circle means “over a closed loop”
The magnetic field is parallel to direction of integration ds
ds cos r d
What if we pick a different path? I
We have demonstrated this is true no matter what path you take
Wire need not even be straight infinite wire
All that matters is that current passes through the closed Ampere loop

4. We will later realize that this formula is imperfect
Understanding Ampere’s Law
If multiple currents flow through, add up all that are inside the loop
Use right-hand rule to determine if they count as + or –
Curl fingers in direction of Ampere loop
If thumb points in direction of current, plus, otherwise minus
The wire can be bent, the loop can be any shape, even non-planar
5 A
2 A
1 A
4 A
7 A
We will later realize that this formula is imperfect

5. Using Ampere’s Law
Ampere’s Law can be used – rarely – to calculate magnetic fields
Need lots of symmetry – usually cylindrical
A wire of radius a has total current I distribu-ted uniformly across its cross-sectional area. Find the magnetic field everywhere. I
End-on view
Draw an Ampere loop outside the wire – it contains all the current
Magnetic field is parallel to the direction of this loop, and constant around it
Use Ampere’s Law
But we used a loop outside the wire, so we only have it for r > a

6. Using Ampere’s Law (2) Now do it inside the wire
Ampere loop inside the wire does not contain all the current
The fraction is proportional to the area a
End-on view

7. Field Inside a Solenoid
It remains only to calculate the magnetic field inside
We use Ampere’s law
Recall, no significant B-field outside
Only the inside segment contributes
There may be many (N) current loops within this Ampere loop
Let n = N/L be loops per unit length L
Works for any shape solenoid, not just cylindrical
For finite length solenoids, there are “end effects”
Real solenoids have each loop connected to the next, like a helix, so it’s just one long wire

8. A Tesla meter2 is also called a Weber (Wb)
Magnetic Flux
Magnetic flux is defined exactly the same way for magnetism as it was for electricity
A cylindrical solenoid of radius 10 cm has length 50 cm and has 1000 turns of wire going around it. What is the magnetic field inside it, and the magnetic flux through it, when a current of 2.00 A is passing through the wire?
A Tesla meter2 is also called a Weber (Wb)

9. Electromotive Force (EMF)
Faraday’s Law
Electromotive Force (EMF)
Suppose we have some source of force on charges that transport them
Suppose it is capable of doing work W on each charge
It will keep transporting them until the work required is as big as the work it can do + – q
The voltage difference at this point is the electromotive force (EMF)
Denoted E

10. Right hand rule for Faraday’s Law:
Motional EMF
Suppose you have the following circuit in the presence of a B-field
Charges inside the cylinder
Now let cylinder move
Moving charges inside conductor feel force
Force transport charges – it is capable of doing work
This force is like a battery - it produces EMF
v  B W v L B
v is the rate of change of the width W
We can relate this to the change in magnetic flux
Right hand rule for Faraday’s Law:
EMF you get is right-handed compared to direction you calculated the flux

11. Power and Motional EMF Resistor feels a voltage – current flows v L R
Where does the power come from?
Current is in a magnetic field B
To get it to move, you must oppose this force
You are doing work
The power dissipated in the resistor matches the mechanical power you must put in to move the rod

12. How to Make an AC Generator
Have a background source of magnetic fields, like permanent magnets
Add a loop of wire, attached to an axle that can be rotated
Add “commutators” that connect the rotating loop to outside wires
Rotate the loop at angular frequency 
Magnetic flux changes with time
This produces EMF
To improve it, make the loop repeat many (N) times A

13. Sample Problem
A rectangular loop of wire 20 cm by 20 cm with 50 turns is rotated rapidly in a magnetic field B, so that the loop makes 60 full rotations a second. At t = 0 the loop is perpendicular to B. (a) What is the EMF generated by the loop, in terms of B at time t? (b) What B-field do we need to get a maximum voltage of 170 V?
The angle is changing constantly with time
After 1/60 second, it must have gone in one full circle 
loop of wire
The flux is given by:
The EMF is given by:

14. Comments on Generators
The EMF generated is sinusoidal in nature (with simple designs)
This is called alternating current - it is simple to produce
This is actually how power is generated
Generators extremely similar to motors – often you can use a single one for both
Turn the axle – power is generated
Feed power in – the axle turns
Regenerative braking for electric or hybrid cars
Generators:
When current does not flow, there is little resistance to turning the axle
When current does flow, magnetic fields produce forces that resist turning the axle
Motors:
When power is demanded, they require a lot of electric current
When power is not needed, little power keeps them going

15. Ground Fault Circuit Interruptors
Fuses/circuit breakers don’t keep you from getting electrocuted
But GFI’s (or GFCI’s) do
GFCI
Under normal use, the current on the live wire matches the current on the neutral wire
Ampere’s Law tells you there is no B-field around the orange donut shape
Now, imagine you touch the live wire – current path changes (for the worse)
There is magnetic field around the donut
Changing magnetic field means EMF in coiled wire
Current flows in coiled wire
Magnetic field produced by solenoid
Switch is magnetically turned off

16. Inductance Self-Inductance A
Consider a solenoid L, connect it to a battery
Area A, lengthl, N turns
What happens as you close the switch?
Lenz’s law – loop resists change in magnetic field
Magnetic field is caused by the current
“Inductor” resists change in current A l + – E

17. Inductors An inductor in a circuit is denoted by this symbol: L
An inductor satisfies the formula:
L is the inductance
Measured in Henrys (H) L
Kirchoff’s rules for Inductors:
Assign currents to every path, as usual
Kirchoff’s first law is unchanged
The voltage change for an inductor is L (dI/dt)
Negative if with the current
Positive if against the current
In steady state (dI/dt = 0) an inductor is a wire I + – L E

18. Energy in Inductors Is the battery doing work on the inductor? L+ – L E
Integral of power is work done on the inductor
It makes sense to say there is no energy in inductor with no current
Energy density inside a solenoid?
Just like with electric fields, we can associate the energy with the magnetic fields, not the current carrying wires

19. RL Circuits An RL circuit has resistors and inductors
Suppose initial current I0 before you open the switch
What happens after you open the switch?
Use Kirchoff’s Law on loop
Integrate both sides of the equation R I L + –

20. RL Circuits (2) Where did the energy in the inductor go?
How much power was fed to the resistor?
Integrate to get total energy dissipated
It went to the resistor
Powering up an inductor:
Similar calculation L E + – R

21. Sample Problem
An inductor with inductance 4.0 mH is discharging through a resistor of resistance R. If, in 1.2 ms, it dissipates half its energy, what is R? I
L = 4.0 mH R

22. Inductors in Series Inductors in Parallel L2 L1
For inductors in series, the inductors have the same current
Their EMF’s add
Inductors in Parallel
For inductors in parallel, the inductors have the same EMF but different currents L1 L2

23. Parallel and Series - Formulas
Capacitor
Resistor
Inductor
Series
Parallel
Fundamental Formula