wed. (c 17) 5.1.3 lorentz force law: currents thurs. hw6...
TRANSCRIPT
Wed. Thurs. Fri.
(C 17) 5.1.3 Lorentz Force Law: currents (C 17) 5.2 Biot-Savart Law
HW6
Mon. Tues. Wed. Fri.
(C 17) 5.2 Biot-Savart Law T5 Quiver Plots (C 21.6-7,.9) 1.3.4-1.3.5, 1.5.2-1.5.3, 5.3.1-.3.2 Div & Curl B (C 21.6-7,.9) 5.3.3-.3.4 Applications of Ampere’s Law
HW7
Magnetic Force on Charge Distribution 1-D collection of moving charges: wire
BvdqF
Bvdl
dqdl
Bvdl
BIdl
vI
For charge flow confined to infinitesimally-thin wire must flow along wire, so and ldv ˆˆ BIldBIdlF
Note: I and B can vary along (be functions of) length.
B
vI
2-D collection of moving charges: surface B
BvdqF
Bvda
dqda
Bvda
vK
Surface Current Density
dw
IdK
dw
BKda
3-D collection of moving charges: volume
BvdqF
Bvd
dqd
Bvd
BJd
Volume Current Density vJ
da
IdJ
daB
Exercise: a) Current I flows down wire of radius a. If it’s uniformly distributed over the wire’s cylindrical surface, then what is the surface current density K?
b) Current I flows down wire of radius a. If it’s distributed throughout the volume such that
𝐽 ∝1
𝑠 , what would be the expression for the volume current density?
Charge Continuity Equation 3-D collection of moving charges
Volume Current Density vJ
da
IdJ
da
Current crossing through closed surface is rate of change of charge in enclosed volume:
IadJ
Fundamental theorem for Divergences dJ
dt
dq
ddt
d I
Equating Integrands J
dt
d
A divergence of current means a depletion of charge
Biot-Savart Law Suggested
ve' = electron velocity
measured by charge
Charge’s frame:
q
-vq lab
q
x
e
' '
'
'
qe
e
'' wirewireq EqF
q
q
wire vI
EEE o
r
'
4
2'''
Define and 2
1
coo
evI
'4
2' qqwireq
IqvF o
r
yI
qvF o
qwireqˆ
24r
Lab’s frame:
= electron
= ionic atomic core
ve = electron velocity measured by us in the “lab frame”
- =+ = electrons’ charge density
(coulombs/meter) labx
e
q vq
r
r24
Ivq o
r
r24
IB o
24
ˆ
r
r
ldIrBd o
Generally for a morsel of wire:
(we’ve not yet proven)
Biot-Savart Law
24
ˆ
r
r
ldIrBd o
I
r
r
Source
Observation location
dL
r
See 17_Bwire_with_r.py, 17_B_long_wire.py
Biot-Savart Law
24
ˆ
r
r
ldIrBd o
I
r
r
Observation location
dl
r
Example 5.5: Field of Infinite Wire (connecting to result of relativistic argument)
x
z
Iˆ
142/12
0
2
0 ˆ4 r
r
dIrB
3
0
4 rr
dI
y zy
yyzzyydI
2/322
0ˆˆˆ
4
x
zy
zydI
y
ˆ4
2/322
0
xd
z
zI
z
y
z
y
z
y
ˆ
142/322
0
x
d
z
Iˆ
142/32
0
x
z
Iˆ
1142/122/12
0
x
z
Iˆ
12
42/12
0
x
z
Iˆ
1
12
42/11
0
2
x
z
Iˆ
2
4
0
Biot-Savart Law
24
ˆ
r
r
ldIrBd o
I
r
r
Observation location
dl
r
Example 5.5: Field of Infinite Wire (Book’s approach)
2
0 ˆ4 r
r
dIrB
xz
Iˆ
2
4
0
a
atanzr
aa
a
dzdl
zdrddl
2cos
1
tan
acos
zr aa cos90sinˆˆ
ddd rr
xdd ˆˆˆ rr
xd
IrB ˆcos
4 2
0
ra
x
dz
IrB ˆcos
cos
1
4 2
20
r
aaa
xz
zdIrB ˆ
cos
cos
4 2
2
0
a
aa
xrd
I ˆcos4 2
0
a
a
r
x
z
dI ˆ
cos
4
0 aa
xdz
Iˆsin
4
2
2
0
a
B field of loop
x y
z
r
Obs. 343424
ˆ
r
rIR
r
rlI
r
rlIB ooo
Rl
1. Visualize Representative segment’s field
2. Expression for segment’s field
3. Sum up field due to all segments
4. Evaluate results
r
r
B field of loop
x y
z
r
r r
Obs. B
'
1. Visualize Representative segment’s field
2. Expression for segment’s field
3. Sum up field due to all segments
4. Evaluate results
0,cos,sin r
cossin,sin,cos xyrzzr
24
ˆ
r
r
lIo
34r
r
lIo
rr
zryrx ,sin,cos
l
r ˆ r
zryrx
rr
zyx
l
sincos
0cossin
ˆˆˆ
detr
B field of loop
x y
z
r
r
r
Obs. B
'
1. Visualize Representative segment’s field
2. Expression for segment’s field
3. Sum up field due to all segments
4. Evaluate results
cossin,sin,cos xyrzzrl r
34r
r
lIo
rr
zryrx ,sin,cos
l
21
sincos22222 yxrrzyx
21
222sincos zryrx
21
sincos222 yxrrr
23
sincos2
cossin,sin,cos
224
yxrrr
xyrzzrIB o
23
sincos2
cossin,sin,cos
224
yxrrr
xyrzzrIB o
B field of loop
1. Visualize Representative segment’s field
2. Expression for segment’s field
3. Sum up field due to all segments
4. Evaluate results
x y
z
r
Obs.
r
B
l
loop
BB
where
By components
2
022
423
sincos2
cos
yxrrr
dzrIB o
x
2
0
B
2
0
xx BB
2
022
423
sincos2
cos
yxrrr
zrIB o
x
r
23
sincos2
cossin,sin,cos
224
yxrrr
xyrzzrIB o
B field of loop
1. Visualize Representative segment’s field
2. Expression for segment’s field
3. Sum up field due to all segments
4. Evaluate results
x y
z
r
Obs.
r
B
l
loop
BB
where
By components
2
022
423
sincos2
cos
yxrrr
dzrIB o
x
2
0
B
2
022
423
sincos2
sin
yxrrr
dzrIB o
y
2
022
423
sincos2
cossin
yxrrr
dxyrrIB o
z
Similarly
r
2
022
423
sincos2
cossin
yxrrr
dxyrrIB o
z
2
022
423
sincos2
sin
yxrrr
dzrIB o
y
2
022
423
sincos2
cos
yxrrr
dzrIB o
x
B field of loop
1. Visualize Representative segment’s field
2. Expression for segment’s field
3. Sum up field due to all segments
4. Evaluate results
x y
z
r
Obs.
r
B
l
Simple case for point on z-axis: oo zr ,0,0
r
2
022
2
423
rr
drIB o
z
2
022
423
sin
rr
dzrIB o
y
B field of loop
1. Visualize Representative segment’s field
2. Expression for segment’s field
3. Sum up field due to all segments
4. Evaluate results
x y
z
r
Obs.
r
B
l
Simple case for point on z-axis: zr ,0,0
0cos
2
022
423
d
rz
zrIo
2
022
423
cos
rr
dzrIB o
x
0sin
2
022
423
d
rz
zrIo
2
022
2
423
drz
zrIo
2
23
22
2
4
rz
rIo
r
B field of Solenoid Simple case for point on z-axis: zr ,0,0
2
23
22
2
4
rzz
rIdB o
z
r
z'
z
N rings over length L, or z
N
L
Nn
ringsall
zz dBB.
ringsall
z
rzz
znrIB o
.22
2
42
23
top
bottom
o
z
zz
z
rzz
zdnrIB
23
22
2
2
B field of Solenoid Simple case for point on z-axis: zr ,0,0
r
top
bottom
o
z
zz
z
rzz
zdnrIB
23
22
2
2
a
asin
21
22 rrzz
acotrzz
acotdrzd aa
dr
2sin
top
bottom
odr
rnrIBz
a
a
a
aa23
32
2sin
sin
top
bottom
o dIn
a
a
aasin
2
bottomtopIno aa
sinsin2
Wed. Thurs. Fri.
(C 17) 5.1.3 Lorentz Force Law: currents (C 17) 5.2 Biot-Savart Law
HW6
Mon. Tues. Wed. Fri.
(C 17) 5.2 Biot-Savart Law T5 Quiver Plots (C 21.6-7,.9) 1.3.4-1.3.5, 1.5.2-1.5.3, 5.3.1-.3.2 Div & Curl B (C 21.6-7,.9) 5.3.3-.3.4 Applications of Ampere’s Law
HW7