Step 1: Cut up the distributioninto pieces

B

origin: center of the solenoid

Step 2: Contribution of one piece

Bz 0

42R2I

R2 d z 2 3/2one loop:

Number of loops per meter: N/L

Number of loops in z: (N/L) z

Field due to z: Bz 0

42R2I

R2 d z 2 3/2

N

Lz

Magnetic Field of a Solenoid

Step 3: Add up the contributionof all the pieces

B

dBz 0

42R2I

R2 d z 2 3/2

N

Ldz

Bz 0

42R2NI

L

dz

R2 d z 2 3/2 L /2

L /2

Bz 0

42NI

L

d L / 2

d L / 2 2 R2

d L / 2

d L / 2 2 R2

Magnetic field of a solenoid:

Magnetic Field of a Solenoid

Bz 0

42NI

L

d L / 2

d L / 2 2 R2

d L / 2

d L / 2 2 R2

Special case: R<<L, center of the solenoid:

Bz 0

42NI

L

L / 2

L / 2 2

L / 2

L / 2 2

0

42NI

L2

L

NIBz

0 in the middle of a long solenoid

Magnetic Field of a Solenoid

Triangular coil

𝑟

𝑟

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?

Helmholtz CoilsThere is a current going through the two identical loops producing a magnetic dipole moment of in each loop. Which direction is B on the x-axis?

How what is B near the origin? Assume that the positions of the loops are large compared to their radii.

𝐷 𝑥−𝐷

𝐵𝑙𝑜𝑜𝑝=𝜇0

4𝜋2𝜇𝑧 3

𝐼

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.

Ampere’s law

Quantifying the Magnetic Field Pattern

r

IBwire

2

40

Curly character – introduce: ldB

dlr

IldB

2

40

rr

I

22

40

IldB 0

Similar to Gauss’s law (Q/0)

Will it work for any circular path of radius r ?

IldB 0

Need to compare and11 ldB

22 ldB

||BdlldB

2

||2

1

1

r

dl

r

dl

1

01

2

4 r

IB

12

1

2

02

2

4B

r

r

r

IB

1

1

21

2

1||2222 dl

r

rB

r

rdlBldB

1122 ldBldB

A Noncircular Path

𝑑𝑙2∥=𝑟2

𝑟1

𝑑𝑙1

Where in loop doesn’t matter!

Currents Outside the Path

IldB 0

Need to compare and11 ldB

22 ldB

2

||2

1

1

r

dl

r

dl 1

2

12 B

r

rB

1122 ldBldB

0 ldB

for currents outside the path

101 IldB

202 IldB

03 ldB

pathinsideIldB _0

Ampere’s law

Three Current-Carrying Wires

∮ (𝐵1+𝐵2+𝐵3 ) ∘𝑑 𝑙=𝜇0 ( 𝐼1− 𝐼 2 )

All the currents in the universe contribute to Bbut only ones inside the path result in nonzero path integral

Ampere’s law is almost equivalent to the Biot-Savart law:but Ampere’s law is relativistically correct

Ampère’s Law

pathinsideIldB _0

1. Choose the closed path2. Imagine surface (‘soap film’) over the path

ldB

3. Walk counterclockwise around the path adding up

4. Count upward currents as positive, inward going as negative

21_ III pathinside uppathinside II _ updownup III

Inside the Path

pathinsideIldB _0

Ampere’s law

What is Bd

rl— ?

A) 0 TmB) 8.7 Tm C) 1.7 TmD) 2.0 Tm E) 2.1 Tm

= .866

, , w=0.5m, h=0.2m,

What is ?

0

4110 7 T m

A

What is I ?

A) AB) AC) AD) A

pathinsideIldB _0

pathinsideIldB _0

Can B have an out of plane component?

Is it always parallel to the path?

rBldB 2

IrB 02

r

IB

2

40

for thick wire: (the same as for thin wire)

Would be hard to derive using Biot-Savart law

Ampere’s Law: A Long Thick Wire

pathinsideIldB _0

Number of wires: (N/L)d

What is on sides? ldB

B outside is very small

BdldB

Bd 0I N / L dL

INB 0 (solenoid)

Uniform: same B no matter where is the path

Ampere’s Law: A Solenoid

pathinsideIldB _0

Symmetry: B || path

INrB 02

r

NIB

2

40

Is magnetic field constant acrossthe toroid?

Ampere’s Law: A Toroid