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Lecture 12
Physics 1202: Lecture 12
Today s Agenda
•  Announcements:
–  Lectures posted on:
www.phys.uconn.edu/~rcote/
–  HW assignments, solutions etc.
•  Homework #4:
–  Not this week ! (time to prepare midterm)
•  Midterm 1:
–  Friday Oct. 2
–  Chaps. 15, 16 & 17.
x x x x x x x x x x x x
x x x x x x x x x x x v
x B
x x x x x x x x x x x x
v
F
F q
S
N
1
Lecture 12
Magnetic Force on a Current
� ×B
�
F� = I L
or
|F� | = ILB sin θ
Current loop &
Magnetic Dipole Moment
•  No net force
•  If plane of loop is not ⊥ to field, there
will be a non-zero torque on the loop!
θ
•  We can define the magnetic dipole
moment of a current loop as follows:
magnitude:
µ=AI
B
x
w
F
θ
F
.
µ
direction: right-hand rule
•  Torque on loop can then be rewritten as:
τ = A I B sinθ
⇒
•  Note: if loop consists of N turns, µ = N A I
2
Lecture 12
Calculation of Magnetic Field
•  Two ways to calculate the Magnetic Field:
•  Biot-Savart Law:
×
I
"Brute force"
•  Ampere's Law
"High symmetry"
•  These are the analogous equations for the Magnetic Field!
µ0= 4π X 10-7 T m /A: permeability (vacuum)
Magnetic Field of ∞ Straight Wire
∴
Direction of B:
right-hand rule
3
Lecture 12
Lecture 12, ACT 1
•  I have two wires, labeled 1 and 2, carrying equal
current, into the page. We know that wire 1
produces a magnetic field, and that wire 2 has
moving charges. What is the force on wire 2 from
wire 1 ?
(a) Force to the right
Wire 1
Wire 2
X
X
I
I
(b) Force to the left (c) Force = 0
Force between two conductors
•  Force on wire 2 due to B at wire 1:
•  Force on wire 2 due to B at wire 1:
•  Total force between wires 1 and 2:
•  Direction:
attractive for I1, I2 same direction
repulsive for I1, I2 opposite direction
4
Lecture 12
Circular Loop
•  Circular loop of radius R
carries current i. Calculate B
along the axis of the loop:
>
•
R θ
I
ΔB
θ
z
>
•  Symmetry ⇒ B in z-direction.
r
R
r
ΔB
x
⇒
•  At the center (z=0):
Bz =
µ0 I
2R
Bz =
Lecture 12, ACT 2
(a) Bz(A) < 0
(b) Bz(A) = 0
N µ0 I
2R
for N coils
•  Note the form the field takes for z>>R:
•  Equal currents I flow in identical
circular loops as shown in the
diagram. The loop on the right (left)
carries current in the ccw (cw)
direction as seen looking along the
+z direction.
–  What is the magnetic field Bz(A)
at point A, the midpoint between
the two loops?
z
I
o
I
x
B
A
x
z
o
(c) Bz(A) > 0
5
Lecture 12
Lecture 12, ACT 2
•  Equal currents I flow in identical
circular loops as shown in the
diagram. The loop on the right
(left) carries current in the ccw
(cw) direction as seen looking
along the +z direction.
I
o
I
x
B
A
x
z
o
–  What is the magnetic field Bz(B) at point B, just to the right of
the right loop?
(a) Bz(B) < 0
(b) Bz(B) = 0
(c) Bz(B) > 0
6
Lecture 12
B Field of a
Solenoid
•  A constant magnetic field can (in principle) be produced by an
∞ sheet of current. In practice, however, a constant magnetic
field is often produced by a solenoid.
L
•  A solenoid is defined by a current I flowing
through a wire which is wrapped n turns per
unit length on a cylinder of radius a and
length L.
a
•  If a << L, the B field is to first order contained within the
solenoid, in the axial direction, and of constant magnitude.
In this limit, we can calculate the field using Ampere's Law.
7
Lecture 12
B Field of a
∞ Solenoid
•  To calculate the B field of the ∞ solenoid using Ampere's Law,
we need to justify the claim that the B field is 0 outside the
solenoid.
•  To do this, view the ∞ solenoid from the
side as 2 ∞ current sheets.
•  The fields are in the same direction in the
region between the sheets (inside the
solenoid) and cancel outside the sheets
(outside the solenoid).
⇒
xxxxx
• •• • •
(n: number of
turns per unit
length)
8
Lecture 12
Toroid
•
•  Toroid defined by N total turns with
current i.
•
•  B=0 outside toroid!
•
x
x
x
x
x
•
•  B inside the toroid.
•
•
•
•
•
•
xx x
x x
•
x
x
r xx
xx
• B•
•
•
•
⇒
9
Lecture 12
Magnetism in Matter
•  When a substance is placed in an external magnetic field Bo,
the total magnetic field B is a combination of Bo and field due
to magnetic moments (Magnetization; M):
– 
B = Bo + µoM = µo (H +M) = µo (H + χ H) = µo (1+χ) H
»  where H is magnetic field strength
•  χ is magnetic susceptibility
•  Alternatively, total magnetic field B can be expressed as:
–  B = µm H
»  where µm is magnetic permeability
»  µm = µo (1 + χ )
•  All the matter can be classified in terms of their response to
applied magnetic field:
–  Paramagnets
–  Diamagnets
–  Ferromagnets
µm > µo
µm < µo
µm >>> µo
Faraday's Law
n
B
B
N
θ
B
S
v
S
N
B
v
! !
! B = B • A = BA cos!
"# B
! =!
"t
10
Lecture 12
Induction Effects
•  Bar magnet moves through coil
S
N
N
S
N
S
v
⇒ Current induced in coil
•  Change pole that enters
⇒ Induced current changes sign
•  Bar magnet stationary inside coil
v
⇒ No current induced in coil
v
•  Coil moves past fixed bar magnet
S
⇒ Current induced in coil
N
Faraday's Law
•  Define the flux of the magnetic field B through a surface
A=An from:
! !
! B = B • A = BA cos!
n
B
θ
B
•  Faraday's Law:
The emf induced around a closed circuit is determined by
the time rate of change of the magnetic flux through that
circuit.
! =!
"# B
"t
The minus sign indicates direction of induced current
(given by Lenz's Law).
11
Lecture 12
Faraday
s law for many loops
•  Circuit consists of N loops:
all same area
ΦB magn. flux through one loop
loops in series
emfs add!
! = !N
•  Lenz's Law:
"# B
"t
Lenz's Law
The induced current will appear in such a direction that it
opposes the change in flux that produced it.
S
N
B
v
N
B
S
v
•  Conservation of energy considerations:
Claim: Direction of induced current must be so as to
oppose the change; otherwise conservation of energy
would be violated.
»  Why???
•  If current reinforced the change, then the change
would get bigger and that would in turn induce a
larger current which would increase the change,
etc..
12
Lecture 12
Lecture 12, ACT 3
y
•  A conducting rectangular loop
moves with constant velocity v in
the +x direction through a region of
constant magnetic field B in the -z
direction as shown.
–  What is the direction of the
induced current in the loop?
(a) ccw
(b) cw
XXXXXXXXXXXX
XXXXXXXXXXXX
X X X X X X X vX X X X X
XXXXXXXXXXXX
x
(c) no induced current
Lecture 12, ACT 4
y
• A conducting rectangular loop
moves with constant velocity v in the
-y direction away from a wire with a
constant current I as shown.
•  What is the direction of the
induced current in the loop?
(a) ccw
(b) cw
I
v
x
(c) no induced current
13