Download Lecture 12 - UConn Physics

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.
Magnetic Force on a Current
or
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!
q
• We can define the magnetic dipole
moment of a current loop as follows:
magnitude:
m=AI
direction: right-hand rule
• Torque on loop can then be rewritten as:
t = A I B sinq 
• Note: if loop consists of N turns, m = N A I
B
x
w
F
m
q
F
.
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!
m0= 4 X 10-7 T m /A: permeability (vacuum)
Magnetic Field of  Straight Wire
\
Direction of B:
right-hand rule
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
(b) Force to the left (c) Force = 0
I
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
Circular Loop
• Circular loop of radius R
carries current i. Calculate B
along the axis of the loop:
I
R

• At the center (z=0):
r
z
r
x
Bz =
• Note the form the field takes for z>>R:
m0 I
2R
DB
q
z
>
>
• Symmetry  B in z-direction.
•
Rq
Bz =
DB
N m0 I
2R
for N coils
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.
– What is the magnetic field Bz(A)
at point A, the midpoint between
the two loops?
(a) Bz(A) < 0
(b) Bz(A) = 0
I
o
I
x
B
A
x
(c) Bz(A) > 0
o
z
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
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
z
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.
• 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.
a
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)
Toroid
•
• Toroid defined by N total turns with
current i.
•
• B=0 outside toroid!
•

•
•
xx x
x
x
x
x
xx
•
• B inside the toroid.
•
•
•
•
x
•
x
x
r xx
xx
• B•
•
•
•
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 + moM = mo (H +M) = mo (H + c H) = mo (1+c) H
» where H is magnetic field strength
 c is magnetic susceptibility
• Alternatively, total magnetic field B can be expressed as:
– B = mm H
» where mm is magnetic permeability
» mm = mo (1 + c )
• All the matter can be classified in terms of their response to
applied magnetic field:
– Paramagnets
– Diamagnets
– Ferromagnets
mm > mo
mm < mo
mm >>> mo
Faraday's Law
n
B
B
N
q
S
v
S
B
N
v
FB = B· A = BAcosq
DF B
e =Dt
B
Induction Effects
• Bar magnet moves through coil
S
N
N
S
N
S
 Current induced in coil
• Change pole that enters
 Induced current changes sign
• Bar magnet stationary inside coil
 No current induced in coil
• Coil moves past fixed bar magnet
 Current induced in coil
v
S
N
v
v
Faraday's Law
• Define the flux of the magnetic field B through a surface
A=An from:
n
FB = B· A = BAcosq
B
q
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.
DF B
e =Dt
The minus sign indicates direction of induced current
(given by Lenz's Law).
Faraday’’s law for many loops
• Circuit consists of N loops:
all same area
FB magn. flux through one loop
loops in “series”
emfs add!
DF B
e = -N
Dt
• Lenz's Law:
Lenz's Law
The induced current will appear in such a direction that it
opposes the change in flux that produced it.
B
B
S
N
v
N
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..
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
(c) no induced current
x