mon. wed (c 17) 5.1.3 lorentz force law: currents thurs...
TRANSCRIPT
Mon. Wed Thurs. Fri.
(C 17) 12.1.1-.1.2, 12.3.1 E to B; 5.1.1-.1.2 Lorentz Force Law: fields and forces (C 17) 5.1.3 Lorentz Force Law: currents (C 17) 5.2 Biot-Savart Law
HW6
Force between stationary charges (Coulomb’s Law: Eq’n 2.1)
341
4ˆ
r
rr
r 2
o
qQqQF
o
q Q r
Force between moving charges (Eq’n 10.74)
auuvcc
Vauuvc
u
qQF
o
rrr
r
r 2222
3ˆ
4
vcu
r̂q Q r
v
a
V
“The entire theory of classical electrodynamics is contained in that equation…but you see why I preferred to start out with Coulomb’s law.” - Griffiths
Force between moving charges
q Q r
v
a
V
Electric Magnetic
Also depends on observer’s perception of recipient charge’s velocity
Depends on observer’s perception of source charge’s velocity and acceleration
𝐹 𝑄←𝑞=𝑞𝑄
4𝜋𝜀𝑜
r
r∙𝑢3 𝑐2 − 𝑣2 𝑢 + r × 𝑢 × 𝑎 +
𝑉
𝑐× r × 𝑐2 − 𝑣2 𝑢 + r × 𝑢 × 𝑎
𝑢 ≡ 𝑐r − 𝑣
Note: extremely asymmetric between two charges; reciprocity (Newton’s 3rd) does not hold
But who’s moving how fast is all relative
𝐹 𝑄←𝑞
Crash Course in Special Relativity Principle: Laws of Physics should be the same in all inertial frames of reference
Old frame-transformation math:
laws of E&M referenced an absolute speed, not speed relative to preferred frame
Charge moving near stationary magnet feels magnetic force; charge stationary near moving magnet feels…?
Observations:
One of the three must go
Key to new frame-transformation math:
Speed of light is measured to be the same in all reference frames
tvxx youmemeballyouball
youmemeballyouball vvv
with clock Measured in frame at rest with two events defining interval
Crash Course in Special Relativity New frame-transformation math derived – light clocks and meter sticks
h
With meter stick Measured in frame at rest with two locations of defining events.
Agreeing about both c and v means disagreeing about t and L.
cwcw tvL ..
∆𝑡𝑤.𝑐 = 2ℎ
𝑐 𝑑 = 2ℎ 2 + 𝐿𝑤.𝑠
2 = 2ℎ 2 + 𝑣∆𝑡𝑤.𝑠2
= 𝑐∆𝑡𝑤.𝑐2 + 𝑣∆𝑡𝑤.𝑠
2
Lw.s= 𝑣∆𝑡𝑤.𝑠
Lw.c= 𝑣∆𝑡𝑤.𝑐
Lw.c= 𝑣∆𝑡𝑤.𝑠 1 −𝑣
𝑐
2
Lw.c= 𝐿𝑤. 𝑠 1 −𝑣
𝑐
2
𝑑
∆𝑡𝑤.𝑠∆𝑡𝑤.𝑠 = 𝑐∆𝑡𝑤.𝑐
2 + 𝑣∆𝑡𝑤.𝑠2
𝑐∆𝑡𝑤.𝑠 = 𝑐∆𝑡𝑤.𝑐2 + 𝑣∆𝑡𝑤.𝑠
2
𝑐∆𝑡𝑤.𝑠2 = 𝑐∆𝑡𝑤.𝑐
2 + 𝑣∆𝑡𝑤.𝑠2
1 −𝑣
𝑐
2
∆𝑡𝑤.𝑠2 = ∆𝑡𝑤.𝑐
2
∆𝑡𝑤.𝑐 = ∆𝑡𝑤.𝑠 1 −𝑣
𝑐
2
𝐿𝑤. 𝑠 =Lw.c
1−𝑣
𝑐
2 Clock’s “proper” time:
: Stick’s “proper” length =g Lw.c =
1
𝛾∆𝑡𝑤.𝑠
Crash Course in Special Relativity New frame-transformation math derived – light clocks and meter sticks Example
𝐿𝑤. 𝑠 =Lw.c
1−𝑣
𝑐
2 =g Lw.c ∆𝑡𝑤.𝑐 = ∆𝑡𝑤.𝑠 1 −
𝑣
𝑐
2
=1
𝛾∆𝑡𝑤.𝑠
Think of the ladder and barn problem. Say the Farmer’s got hold of the ladder and he’s going to run at the barn.
But in terms of what you in the station would measure,
But to phrase that in terms of what I in the train would measure,
I’m in a train watching a pool game. I see the ball role a distance to the pocket. I see it’s taking time to get there. Meanwhile, the train and I are rolling through the station in the same direction at speed v. You, standing in the station, see all this happen. Classically, you’d imagine that with my yard stick, I’d measure the ball rolling a distance
Crash Course in Special Relativity Galilean Transformation Corrected
"∆𝑥𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒" =1
γ ∆xball.table
∆𝑥𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒
∆𝑡𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒
so, or, ∆𝑥𝑏𝑎𝑙𝑙. 𝑡𝑎𝑏𝑙𝑒 = g ∆xball.station - v ∆𝑡𝑏𝑎𝑙𝑙. 𝑠𝑡𝑎𝑡𝑖𝑜𝑛)
v
"∆𝑥𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒" = ∆𝑥𝑏𝑎𝑙𝑙. 𝑠𝑡𝑎𝑡𝑖𝑜𝑛-v∆𝑡𝑏𝑎𝑙𝑙.𝑠𝑡𝑎𝑡𝑖𝑜𝑛
1
γ ∆xball.table = ∆𝑥𝑏𝑎𝑙𝑙. 𝑠𝑡𝑎𝑡𝑖𝑜𝑛
-v∆𝑡𝑏𝑎𝑙𝑙.𝑠𝑡𝑎𝑡𝑖𝑜𝑛
Then again, I’d imagine with your yardstick you’d measure the ball rolling a distance
"∆𝑥𝑏𝑎𝑙𝑙.𝑠𝑡𝑎𝑡𝑖𝑜𝑛" = ∆𝑥𝑏𝑎𝑙𝑙. 𝑡𝑎𝑏𝑙𝑒+v∆𝑡𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒
"∆𝑥𝑏𝑎𝑙𝑙.𝑠𝑡𝑎𝑡𝑖𝑜𝑛" =1
γ ∆xball.station
so, ∆𝑥𝑏𝑎𝑙𝑙. 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 =
g∆𝑥𝑏𝑎𝑙𝑙. 𝑡𝑎𝑏𝑙𝑒+v∆𝑡𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒)
Consistent only if ∆𝑡𝑏𝑎𝑙𝑙.𝑠𝑡𝑎𝑡𝑖𝑜𝑛 =g∆𝑡𝑏𝑎𝑙𝑙. 𝑡𝑎𝑏𝑙𝑒 +𝑣
𝑐2 ∆𝑥𝑏𝑎𝑙𝑙. 𝑡𝑎𝑏𝑙𝑒)
∆𝑡𝑏𝑎𝑙𝑙. 𝑡𝑎𝑏𝑙𝑒 = g ∆tball.station -
𝑣
𝑐2 ∆𝑥𝑏𝑎𝑙𝑙. 𝑠𝑡𝑎𝑡𝑖𝑜𝑛)
Note: both observers measure distances as ‘proper’ lengths of their sticks, but neither is a equidistant to the two events-neither measures proper time
I’m in a train watching a pool game. I see the ball role a distance to the pocket. I see it’s taking time to get there. Meanwhile, the train and I are rolling through the station in the same direction at speed v. You, standing in the station, see all this happen.
Crash Course in Special Relativity Galilean Transformation Corrected
v
Some algebra later,
𝑣𝑏𝑎𝑙𝑙. 𝑠𝑡𝑎𝑡𝑖𝑜𝑛=
𝑣𝑏𝑎𝑙𝑙. 𝑡𝑎𝑏𝑙𝑒+ v
1 +𝑣𝑣𝑏𝑎𝑙𝑙. 𝑡𝑎𝑏𝑙𝑒
𝑐2
Or renaming v = vtable.station
𝑣𝑏𝑎𝑙𝑙. 𝑠𝑡𝑎𝑡𝑖𝑜𝑛=
𝑣𝑏𝑎𝑙𝑙. 𝑡𝑎𝑏𝑙𝑒+ 𝑣𝑡𝑎𝑏𝑙𝑒. 𝑠𝑡𝑎𝑡𝑖𝑜𝑛
1 +𝑣𝑡𝑎𝑏𝑙𝑒.𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑣𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒
𝑐2
∆𝑥𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒
∆𝑡𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒
𝑣𝑏𝑎𝑙𝑙.𝑠𝑡𝑎𝑡𝑖𝑜𝑛 =∆𝑥𝑏𝑎𝑙𝑙.𝑠𝑡𝑎𝑡𝑖𝑜𝑛
∆𝑡𝑏𝑎𝑙𝑙.𝑠𝑡𝑎𝑡𝑖𝑜𝑛
𝛾 ∆𝑥𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒 + 𝑣∆𝑡𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒
𝛾 ∆𝑡𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒 + 𝑣𝑐2∆𝑥𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒
=
Crash Course in Special Relativity New frame-transformation math derived – light clocks and meter sticks Example
𝑣𝑏𝑎𝑙𝑙. 𝑠𝑡𝑎𝑡𝑖𝑜𝑛=
𝑣𝑏𝑎𝑙𝑙. 𝑡𝑎𝑏𝑙𝑒+ 𝑣𝑡𝑎𝑏𝑙𝑒. 𝑠𝑡𝑎𝑡𝑖𝑜𝑛
1 +𝑣𝑡𝑎𝑏𝑙𝑒.𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑣𝑏𝑎𝑙𝑙.𝑡𝑎𝑏𝑙𝑒
𝑐2
ve = electron velocity measured by us in the “lab frame”
l+ = ions’ charge density
(coulombs/meter)
Transition / Transformation from E to M
= electron = ionic atomic core
l- =l+ = electrons’ charge density
(coulombs/meter)
Metal wire
Lab frame: you and I see an electrically neutral wire (all be it, with the electrons moving)
labx
e
l
labx
e
l
wirewireq EqF
q
Stationary charge
02222
41
41
41
41
rrrr
EEEoooowire
llll
l+ = ions’ charge density
(coulombs/meter)
Transition / Transformation from E to M
= electron = ionic atomic core
ve = electron velocity measured by us in the “lab frame”
l- =l+ = electrons’ charge density
(coulombs/meter)
Lab frame: you and I see an electrically neutral wire (all be it, with the electrons moving)
labx
e
l
labx
e
l
?wireqF
q
Moving charge
vq
Transition / Transformation from E to M
Charge’s frame:
21c
vv
qe
eqe
vvv
Chain of positive atoms moving backwards at So spacing seen by charge is related to their stationary, “proper” separation (as seen in lab) by
qatoms vv
Chain of electrons moving forward at only
q
lab
q
properatom
atom
xxx
gg
.
2
1
c
vx
q
lab
lab
q
x
e
gl '
Similarly, separation in lab frame relates to “proper” separation (seen in electrons’ rest frame)
labepropere xx g.2
1
1
c
ve
eg
Or charge density appears compressed to
So spacing seen by charge is related to their stationary, “proper” separation (not seen in lab) by
e
propere
e
xx
g
.2
1
1
c
ve
egwhere
where Combined: e
labee
xx
g
g
So, labe
e
x
e
g
gl '' '
'
'
l
gg
g
qe
e
l+’ = ions’ charge density
(coulombs/meter)
Transition / Transformation from E to M
ve' = electron
velocity measured
by charge
l-’ = electrons’ charge density
(coulombs/meter)
Charge’s frame:
q
-vq
lab
q
x
e
gl ' '
'
'
l
gg
g
qe
e
labe
e
x
e
g
gl ''
'' wirewireq EqF
rrEEE
oowire
'''''
ll
224
14
1
rr
qe
e
oo
'' '
' lgg
g
l
22
41
41
'
''
qe
e
ro gg
gl
12
41
21c
vv
qe
eqe
vvv
2
1
1
c
ve
eg 2
1
1
c
ve
egwhere 2
1
1
c
vq
qg
A bit of algebra later, 24
12
c
vv
rE
qeq
wire o
''
gl
'
''
qe
eq
ro gg
ggl
12
41
Define and 2
1
coo
evI l q
q
wire vr
IE o '
'g
24
so
'' qqwireqr
IqvF o g
24
Hand waving: F~ distance/time2
Transformation from rest frame requires factor of g/g21/g r
IqvF o
qwireq
24
Finally, we observe:
𝐹 𝑄←𝑞=𝑞𝑄
4𝜋𝜀𝑜
r
r∙𝑢3 𝑐2 − 𝑣2 𝑢 + r × 𝑢 × 𝑎 +
𝑉
𝑐× r × 𝑐2 − 𝑣2 𝑢 + r × 𝑢 × 𝑎
Jumping in to Magnetism
Electric Magnetic
Also depends on observer’s perception of recipient charge’s velocity
Depends on observer’s perception of source charge’s velocity and acceleration
𝑢 ≡ 𝑐r − 𝑣
𝐹 𝑄←𝑞.𝑚𝑎𝑔=𝑄𝑉 ×𝑞
4𝜋𝑐𝜀𝑜
r
r∙𝑢3 r × 𝑐2 − 𝑣2 𝑢 + r × 𝑢 × 𝑎
Magnetic Field,B
𝐹 𝑄←𝑞.𝑚𝑎𝑔=𝑄𝑉 × 𝐵
Magnetic force
BvEqdt
vdm
dt
pd
xyyxz
zxxzy
yzzyx
zyx
zyx
z
y
x
dtdv
dt
dv
dtdv
BvBvE
BvBvE
BvBvE
q
BBB
vvv
zyx
E
E
E
qm
z
y
x ˆˆˆ
Cyclotron Motion in a Uniform Magnetic Field BvEqF
yz v
m
qB
m
qE
dt
dv
zz v
m
qB
dt
vd2
2
2
m
qB
In general
𝑥
𝑦
𝑧
B
q
v
Book’s Example
E
dt
dv
m
qB
dt
vdzy
2
2Take next derivative
y
yv
m
qB
m
qE
m
qB
dt
vd2
2
z
yv
m
qB
dt
dv
dt
dv
m
qB
dt
vd yz 2
2
Cross substitute
y
yv
m
qB
m
BEq
dt
vd2
2
2
2
2
Guess Solution Forms
tCtCtvz sincos 21
B
EtCtCtvy sincos 43
Plug in
z
yv
m
qB
dt
dv
tCtCm
qBtCtC sincoscossin 2143
Compare terms and conclude 23 CC 14 CC
Cyclotron Motion in a Uniform Magnetic Field BvEqF
m
qB
𝑥
𝑦
𝑧
B
q
v
Book’s Example
E
tCtCtvz sincos 21 B
EtCtCtvy cossin 21
where
For position components, integrate
521 Ct
Ct
Ctz
cossin 6
21 sincos CtB
Et
Ct
Cty
Impose Initial Conditions
00 52 C
Cz
Start at origin 00 6
1 CC
y
2
5
CC
1
6
CC
tC
tC
tz
cossin 121 tB
Et
Ct
Cty
sincos1 21
Cyclotron Motion in a Uniform Magnetic Field BvEqF
m
qB
𝑥
𝑦
𝑧
B
q
v
Book’s Example
E
tCtCtvz sincos 21 B
EtCtCtvy cossin 21
where
tC
tC
tz
cossin 121 tB
Et
Ct
Cty
sincos1 21
00 v
Initial Velocity:
Impose Initial Conditions Initial position: (0,0)
0sin0cos0 21 CCvz B
ECCvy 0cos0sin0 21
B
ECvy 20 0
10 Cvz 0
B
EC 2
So.
tB
Etz
cos1 tt
B
Ety
sin
tB
Etvz sin t
B
Etvy cos1
Cyclotron Motion in a Uniform Magnetic Field BvEqF
m
qB
𝑥
𝑦
𝑧
B
q
v
Book’s Example
E
tCtCtvz sincos 21
mE
tCtCtvy cossin 21
where
tC
tC
tz
cossin 121 tmE
tC
tC
ty
sincos 21 1
yBEv ˆ/0
Initial Velocity:
Impose Initial Conditions
Initial position: (0,0)
0sin0cos0 21 CCvz
mECCvy 0cos0sin0 21
mECvy 20
B
E
10 Cvz 0
B
EmEC
2
Mon. Wed Thurs. Fri.
(C 17) 12.1.1-.1.2, 12.3.1 E to B; 5.1.1-.1.2 Lorentz Force Law: fields and forces (C 17) 5.1.3 Lorentz Force Law: currents (C 17) 5.2 Biot-Savart Law
HW6
Cyclotron Motion in a Uniform Magnetic Field
F mag F B Qv B
xyyx
zxxz
yzzy
zyx
zyx
dtdv
dt
dv
dtdv
mag
BvBv
BvBv
BvBv
q
BBB
vvv
zyx
qm
Bvqdt
vdm
Fdt
pd
z
y
x ˆˆˆ
𝑥
𝑦
𝑧
B
Q
v
yzx v
m
qB
dt
dv
dt
dv
m
qB
dt
vdxzy
2
2
yzy
vm
qB
dt
vd2
2
2
xzx v
m
qB
dt
vd2
2
2
tCtCtvx sincos)( 43
tCtCm
qBtCtC z sincoscossin 2143
23 CC
For zBB zˆ
xzy
vm
qB
dt
dv
tCtCtvy sincos)( 21
Similarly for vx:
Plug in to
m
qBz14 CC
Cyclotron Motion in a Uniform Magnetic Field
F mag F B Qv B
𝑥
𝑦
𝑧
B
Q
v
tCtCtvx sincos)( 12
tCtCtvy sincos)( 21 m
qBz
Integrate for positions:
dttvdx
t
o
x
x
xo
)(
1cossin 12 tC
tC
xtx o
similarly:
1cossin 21 tC
tC
yty o