The graded exams will be returned next Tuesday, Nov 7. You will have until the next class on...
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Transcript of The graded exams will be returned next Tuesday, Nov 7. You will have until the next class on...
The graded exams will be returned next Tuesday, Nov 7. You will have until the next class on Thursday, Oct 6 to rework the problems you got wrong and receive 50% added credit. Make sure you are in class as you will no have another opportunity to rework the exam. I will be going over the answers in class on Thursday. This will also be your only opportunity to ask for corrections/clarifications on any grading mistakes.
The homework assignment will be on line this afternoon but will not be due until Tuesday, Nov 14. This will give you the opportunity to start work on the problems so that you will not be overloaded with homework and the exam rework next week.
Forces and Torques on the Current Loop
' sin(90 ) cos
F IaB
F IbB IbB
Net force is zero
The Torque is not zero
2 ( / 2)sin ( )( sin )
sin
F b IBa b
IBA
Magnetic Dipole Moment
In general:
Magnetic (dipole) moment
(valid for any orientation and shape)
Potential energy of the loop
Torque wants to rotate the loop so
that its moment is oriented along
I IA
I
U
τ A B n B μ B
μ A
μ B
B
Loops and Coils
sinNIAB
For solenoid
Example: Force and Torque on a Circular Current Loop
( sin ) (cos)d l Rd i Rd j
d F Id l B
( sin ) (cos ) 0
0 0x
i j k
d l B Rd Rd
B
d r dF cos sinr R i R j
j IBA B
€
rr × d
r F =
i j k
Rcosθ Rsinθ 0
0 0 − IBxRdθ cosθ
Example: Force on a Current Loop in Non-Uniform Field
0 0 (only z and y components)B z B y
B j kL L
(1)
(2)
(3)
(4)
d F Id l B
01Component to dl is B=
B yk
L
0 01
02
L B y IB LF i I dy i
L
0 0
0 0
0
i j k
d l B dx
B z B y
L L
2F I dl B 2 0F j IB L
03 2
IB LF i
0
4 0B y
F j I dxL
Magnetic Dipole in a Non-uniform Magnetic Field
( )F B
y
; 0
0 ( )
yB B
Fy y
F along y axis
0
0 ( )
B
y
F opposite to y axis
Magnetic Dipoles and How Magnets Work
The Direct-Current Motor
Magnetic Field of a Moving Charge
02
02
sin
4
4
qvB
r
q v rB
r
Magnetic field of a point charge moving with constant velocity
700 0 2
110 T m/A;
4 c
Example: Force between two moving protonsFind the ratio of electric and magnetic forces on the protons
2
20
1
4E
qF
r
024
qvB k
r
Magnetic field of the lower proton at the position of the top one
( )BF q v B
2 20
24B
q vF j
r
22
0 0 2B
E
F vv
F c
Magnetic Field of Current Element
The Biot-Savart law.
02
02
0 1212
12
12
For element of a (fine) wire:
ˆ
4constant permeability of free space:
For the whole "circuit":
ˆ
4For arbitrary distribution of charge flow:
ˆ(1)(2)
4
ˆ( is fro
I dld
r
I dl
r
dVr
rB
rB
j rB
r m point 1 to point 2)
dflow with velocity v
dQ nqAdl
Magnetic field around a straight wire
wire) thefrom distance theis (where
2sin
4
]sin
;cot ;sin
[
sin
4
:magnitude field For the
0
0
0
2
20
a
a
Id
a
I
dadxax
ar
r
dxIB
Magnetic Field of Two Wires
Field at points on the x-axisto the right of point (3)
0 01 2
02 1 2 2
; ;2 ( ) 2 ( )
( )total
I IB B
x d x d
IdB B B
x d
Magnetic field outside of a conductor pair falls off more rapidly
Magnetic field of a circular arc
€
For the field magnitude at O :
B =μ0I
4πR2ds =∫ μ0I
4πR2Rθ
=μ0Iθ
4πR
Magnetic Field of a Circular Current Loop
30
2/3220
2/322
20
2/3220
220
2
][
)(2)(2
2)(4
cos
4)()(
:axis on the fieldFor
x
Rx
RxRx
IR
RRx
RIRx
dsIxBxB x
Falls off just as the electric field of the electric dipole
Magnetic Field on the Axis of a Coil2
02 2 3/ 2
02 2 3/ 2
2( )
;2 ( )
x
x
NIRB
x R
B NIAx R
The magnetic field of a (small) loop behaves “on the outside” like the electric field of the electric dipole of the same orientation – that’s why “magnetic dipole”.
Magnetic force between two parallel conductors with currents
02
'' 0
1 2
1 2
'0
Magnetic field from conductor 2:
2Magnetic force on conductor 1:
2Absolutely the same magnitude is
for the magnetic force on conductor 2
but
2Currents in the same
B
IB
r
IIF I LB L
r
IIF
L r
F F
direction attract
Currents in opposite directions repel
N/m 102 is
meter 1per force when the
Ampere 1 is
m 1by separated
wiresin twocurrent Identical
:Ampere 1 of Definition
7
Example: Two straight, parallel, superconducting wires 4.5 mm apart carry 15,000 A current each in opposite directions
Should we carry about the mechanical strength of the wires?
'40 10 /
2
IIFN m
L r