1. Magnetic Field of a Solenoid
Step 1: Cut up the distribution
into pieces
Step 2: Contribution of one piece
origin: center of the solenoid
one loop: B
Number of loops per meter: N/L
Number of loops in z: (N/L) z
Field due to z:

2. Magnetic Field of a Solenoid
Step 3: Add up the contribution
of all the pieces B
Magnetic field of a solenoid:

3. Magnetic Field of a Solenoid
Special case: R<<L, center of the solenoid:
in the middle of a long solenoid

4. Triangular coil 𝑟 𝐼
There is a current going through a triangular coil. Which direction is B at the center?
How would you find the magnitude of B?

5. Helmholtz Coils
There is a current going through the two identical loops producing a magnetic dipole moment of 𝜇 in each loop. Which direction is B on the x-axis? 𝐼 −𝐷 𝑥 𝐷
How what is B near the origin? Assume that the positions of the loops are large compared to their radii.
𝐵 𝑙𝑜𝑜𝑝 = 𝜇 0 4𝜋 2𝜇 𝑧 3

6. Patterns of Magnetic Field in Space
Is there current passing
through these regions?
There must be a relationship between the measurements of the magnetic field along a closed path and current flowing through the enclosed area.
Can we predict current pattern front knowing B pattern?
Ampere’s law

7. Quantifying the Magnetic Field Pattern
Curly character – introduce:
Long wire pattern
Similar to Gauss’s law (Q/0)
Will it work for any circular path of radius r ?

8. A Noncircular Path Need to compare and 𝑑 𝑙 2∥ = 𝑟 2 𝑟 1 𝑑 𝑙 1
Long wire pattern
𝑑 𝑙 2∥ = 𝑟 2 𝑟 1 𝑑 𝑙 1
Where in loop doesn’t matter!

9. Currents Outside the Path
Need to compare and
Long wire pattern
for currents outside the path

10. Three Current-Carrying Wires
𝐵 𝐵 𝐵 3 ∘𝑑 𝑙 = 𝜇 0 𝐼 1 − 𝐼 2
Similar to Gauss's law – only charges inside matter
Ampere’s law

11. Ampère’s Law All the currents in the universe contribute to B
but only ones inside the path result in nonzero path integral
Andre Marie Ampere , never attended school (farther educated him). Read Encyclopedia through starting from A etc.
First paper at age 13 – not accepted…
. Maxwell, writing about this Memoir in 1879, says:-
We can scarcely believe that Ampère really discovered the law of action by means of the experiments which he describes. We are led to suspect, what, indeed, he tells us himself, that he discovered the law by some process which he has not shown us, and that when he had afterwards built up a perfect demonstration he removed all traces of the scaffolding by which he had raised it.
Ampere’s law is almost equivalent to the Biot-Savart law:
but Ampere’s law is relativistically correct

12. Inside the Path Ampere’s law Choose the closed path
Imagine surface (‘soap film’) over the path
3. Walk counterclockwise around the path adding up
4. Count upward currents as positive, inward going as negative

13. 𝜃=30°, 𝐵 1 =2×1 0 −4 T, 𝐵 2 =1×1 0 −4 T, w=0.5m, h=0.2m,
What is 𝐵 ∙𝑑 𝑙 ?
What is ?
cos⁡(30°) = .866 c
0 T∙m
8.7 ×1 0 −5 T∙m
1.7 ×1 0 −4 T∙m
2.0 ×1 0 −4 T∙m
2.1 ×1 0 −4 T∙m

14. 𝑟=3cm, 𝐵 =1×1 0 −5 T What is I ? 0 A 1.9×1 0 −1 A 1.9×1 0 −6 A 1.5 AD

15. Ampere’s Law: A Long Thick Wire
Can B have an out of plane component?
Is it always parallel to the path?
Symmetry makes problem easy to solve using Ampere's law – counting currents is simple, and path integral is simple due to the symmetry
for thick wire:
(the same as for thin wire)
Would be hard to derive using Biot-Savart law

16. Ampere’s Law: A Solenoid
Number of wires: (N/L)d
What is on sides?
B outside is very small
Talk importance:
for getting uniform magnetic field
electromagnets
On sides: draw on board (symmetry) to show that B is perp to path on sides (wires to left and right total in only perp component)
:ength d disappears – and it must, B should not depend on some mathematical path
(solenoid)
Uniform: same B no matter where is the path

17. Ampere’s Law: A Toroid Symmetry: B || path
Toroid – B not constant, depends on r, can explain – higher density of wires inside.
Is magnetic field constant across
the toroid?