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Physics 11b
Lecture #12
Sources of the Magnetic Field
S&J Chapter 30
What We Did Last Time
b
„
Lorentz force on current FB = I ∫ ds × B
a
„
„
Simplifies to FB = IL × B if the B field is uniform
Torque on a current loop in a B field τ = IA × B
„
Current loop looks like a magnetic dipole
µ = IA
1
nq
„
RH IB
Hall effect ∆VH =
t
„
Magnetic fields are created by electric current
RH ≡
µ0 Ids × rˆ
„ Biot-Savart Law dB =
4π r 2
µ0 I
„ B field by an infinite straight-line current B =
2π a
Today’s Goals
„
Continue with Biot-Savart law
Calculate B field created by a straight-line current
„ Force between two currents
„
„
Ampère’s Law
Integral form of Biot-Savart law
„ Useful in solving magnetic field problems
„ Examples: thick wire, infinite current sheet, solenoid
„
„
„
Magnetic flux and Gauss’s law of magnetism
Magnetism in matter
Biot-Savart Law
„
B field due to current in a short piece of
wire is given by the Biot-Savart Law
dB
I
µ0 Ids × rˆ
dB =
4π r 2
ds
dB is perpendicular to both ds and r
„ Size of dB depends on the angle between ds and r
„
Ids
sin θ
2
r
Ids
r2
θ
I
ds
dB = 0
r
Straight-Line Current
„
Current I flows in an infinitely-long straight wire
„ B field at distance a from the wire?
y
µ0 Ids × rˆ µ0 Idx sin θ
dB
=
zˆ
dB =
2
2
4π r
4π
r
„ We move ds from x = −∞ to +∞
r
a
„ r and θ changes as we go
θ
r 2 = a2 + x2
a
a
sin θ =
tan θ = −
r
x
„
x
I
z
ds
We can integrate with x, r, or θ Æ which one is the easiest?
Straight-Line Current
„
Easiest to integrate if switch from dx to dθ
adθ
a
y
x=−
dx =
tan θ
sin 2 θ
dB
a
„ Also r =
sin θ
a
µ0 Idx sin θ
dB =
4π
r2
µ0 I sin θ dθ
I
=
z
a
4π
µ0 I π
µ0 I
µ0 I
π
sin θ dθ =
B=
[ − cos θ ]0 =
∫
0
4π a
4π a
2π a
r
θ
x
ds
B field goes around
the wire and
decrease as 1/a
B Field Around Current
„
B field circles around (infinitely-long) current
Direction of B follows right-hand rule
„ Magnitude B decreases as
„
„
µ0 I
B=
2π r
cf. E field created by infinitely-long
linear charge distribution was
λ
charge
E=
density (C/m)
2πε 0 r
„
Similar, but E points outward, B rotates around
B
r
I
Force Between Currents
„
Run two straight wires in parallel
„ I1
„ I2
µ0 I1
B=
2π a
F
feels the force
a
Parallel currents attract each other
„
„
I2
creates rotating B field
µ0 I1 I 2 A
F = I 2L × B =
2π a
„
I1
If I1 and I2 flow in opposite directions, they repel each other
For I1 = I 2 = 1A and a = A = 1m
µ0 (1A) 2 (1m)
F=
= 2 × 10−7 N
2π (1m)
This is in fact how
Ampere is defined
A
Ampère’s Law
„
µ0 I
The denominator of B =
looks like the circumference
2π r
of a circle
„
ds
If we draw a circle around the
current and line-integrate
r
µ0 I
v∫ B ⋅ ds = v∫ 2π r ds = µ0 I
„
This is generalized to Ampère’s Law
For any closed path, the line integral of the magnetic field is
v∫ B ⋅ ds = µ I
0
where I is the total current passing through any surface
bounded by the closed path
Ampère’s Law
„
Ampère’s Law follows Biot-Savart Law
Same way as Gauss’s Law follows Coulomb’s Law
„ It’s useful when a symmetry of the problem helps us to
predict at least the direction of the B field
„
„
Example: long wire with finite thickness
„
Current I flows uniformly through a cylindrical wire
R
„
I
What’s the B field outside and inside of the wire?
Thick Wire
„
Inside the wire
Circle with radius r1 < R encircles
current π r 2 Ir 2
I 1 2 = 12
πR
R
„ Ampère’s Law
µ0 I
Ir12
B=
r
2 1
v∫ B ⋅ ds = 2π r1B = µ0 R 2
2π R
„
„
R
r1
r2
Outside the wire
Circle with radius r2 > R encircles I
„ Ampère’s Law
µ0 I
=
B
v∫ B ⋅ ds = 2π r2 B = µ0 I
2π r2
„
Same result as a
thin wire
Thick Wire
„
Combining, we find
⎧ µ0 I
⎪⎪ 2π R 2 r r ≤ R
B=⎨
⎪ µ0 I
r>R
⎪⎩ 2π r
R
r1
B(r )
r2
µ0 I
2π R
B∝
B∝r
R
1
r
This part is same
as thin wires
r
Infinite Current Sheet
„
Infinite sheet carries current density Js A/m
B fields are parallel to the sheet
opposite directions above
and below
„ Draw a rectangle and
apply Ampère
„
v∫ B ⋅ ds = 2BA = µ I A
0 s
B
B=
„
µ0 I s
2
Uniform B field is created
B
A
Two Infinite Current Sheets
„
What if we had two current sheets?
B=
Each sheet makes uniform B
fields above and below
„ Add up Æ Uniform B field only
between the sheets
„
„
2
This looks similar to a capacitor
Charged sheet makes E fields
above and below
„ Two oppositely-charged sheets
make uniform E field between
„
„
µ0 I s
Infinite current sheet is less
practical to make…
B=0
B = µ0 I s
B=0
Solenoid
„
Bend the infinite current sheet into a tube
Current flows around the tube
„ Uniform B field inside?
„
„
We can make this by winding
a long wire around a cylinder
„
Suppose the winding is dense
i.e. large
N
n=
A
Turns of winding
Solenoid length
Also suppose the solenoid is very long
„ Use Ampère’s Law to find out B
„
Solenoid Field
„
B field is parallel to the solenoid’s axis
Also, if you go far away, B Æ 0
„ Apply Ampère to the rectangular path
„
B=0
v∫ B ⋅ ds = BA = µ InA
0
B = µ0 nI
„
Line integral doesn’t change
as long as the top and bottom
sides of the rectangle remain
outside and inside
Æ B field is uniform
B
A
Magnetic Flux
„
We can define magnetic flux just like the electric flux
Φ B = ∫ B ⋅ dA
Φ E = ∫ E ⋅ dA
Integral is taken over a given surface area
„ “How many magnetic field lines go through this area?”
„
„
Units are
T·m2 = N·m/A for magnetic flux
„ V·m for electric flux
„
„
Main use of the electric flux was Gauss’s Law
„
What happens to it with magnetic field?
Gauss’s Law
„
With electric flux, we had Φ E = v∫ E ⋅ dA =
„
„
qin
ε0
But there is no magnetic “charge”
Gauss’s Law in magnetism:
Φ B = v∫ B ⋅ dA = 0
i.e., magnetic flux through any closed surface is zero
„ Not as useful (in solving problems) as the electric version
„
„
„
But we’ve got Ampère’s Law instead
Magnetic flux is a critical ingredient for Faraday’s Law
„
We’ll get to that next week
Magnetism in Matter
„
Different materials react differently to magnetic field
„
„
Read textbook 30.8
„
„
Some (e.g. iron) stick and others don’t
Things get a bit encylopedic
Three types of magnetism
„
Ferromagnetism – strongly attracted to magnets
„
„
Paramagnetism – weakly attracted to magnets
„
„
Iron (ferrrum), cobalt, nickel, etc.
Ferromagnetic metals turn paramagnetic at high temperature
Diamagnetism – weakly repelled by magnets
„
„
Majority of materials
Special case: superconductors are strongly diamagnetic
Summary
„
Applied Biot-Savart law to linear current B =
µ0 I
2π r
B field rotates around the current
µ0 I1 I 2 A
„ Parallel wires attract each other by F =
„
„
„
Ampère’s Law
„
2π a
Definition of Ampere
v∫ B ⋅ ds = µ I
0
Applies to any boundary that encircle current I
Examples: thick wire, infinite current sheet
„ Solenoid field B = µ0 nI
„
v∫ B ⋅ dA = 0
„
Gauss’s Law Φ B =
„
Ferro-/para-/diamagnetism in matter