Lecture 3-1 Dipole in uniform electric fields No net force. The electrostatic forces on the...
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Transcript of Lecture 3-1 Dipole in uniform electric fields No net force. The electrostatic forces on the...
Lecture 3-1
Dipole in uniform electric fields
• No net force. The electrostatic forces on the constituent point charges are of the same magnitude but along opposite directions. So, there is no net force on the dipole and thus its center of mass should not accelerate.
• Net torque! There is clearly a net torque acting on the dipole with respect to its center of mass, since the forces are not aligned.
http://qbx6.ltu.edu/s_schneider/physlets/main/dipole_torque.shtml
Ep
Lecture 3-2 Torque on the Dipole
The magnitude of the torque:
sin
sinsin
sin)(sin
pE
qEdFd
xdFFx
The torque points into the screen.
Ep
x
F
F
CM
Fr
Lecture 3-3
Electric Field from Coulomb’s Law
20
1ˆ
4i
ii i
qE r
r
++
+ +
+
--
-+-
-
Bunch of Charges
dV
dq dA
dL
(volume charge)
(surface charge)
(line charge)
Continuous Charge Distribution
dq2
0
1ˆ
4
dqE r d E
r
��������������P
r
Summation over discrete charges
Integral over continuous charge distribution
Pir
iq
k
http://www.falstad.com/vector3de/
Lecture 3-4
Continuous Charge Distribution 1: Charged Line
/ 2
2/ 2
/ 2
2/ 2
/ 2
2
2
/
2 2
( )
1
( /
/ /
2)
p
p
p
p
L
x Lp
x L
px L
x L
x L
p
p p
dxE E k
x x
duk u x x
u
k
x
u
L k
k L
x L
Q x
At a point P on axis:
Q
L
/ 2px L
Lecture 3-5
Again: Continuous Charge Distribution 1: Charged Line
At a point P on perpendicular axis:
/ 2
2 2/ 2
/ 2
2 2 3/ 2/ 2
2 2
2 2 2 3/ 2
cos
tan( )
sec
( tan )
cos (2sin )
/ 2 2 /
L
y L
L
L
dxE E k
x y
yk dx x y
x y
yk d
y y
k
L and E k y
kd
y y
1 2
x
Lecture 3-6Physics 241 –Warm-up quiz
The rod is uniformly charged with a positive charge density . What is the direction of the electric field at a point P on a line perpendicular to the rod? Note that the line and the rod are in the same plane.
a) to the right b) to the leftc upd) downe) lower right
p
Lecture 3-7General location of P: Charged Line
2 2
1 1
2
1
2 2 2 3/ 2
2 12 2 1/ 2
2 1
sin( )
1 cos cos
( )
cos cos
x x
x x x
x
x
dx xE k k dx
r x y
k kx y y y
k
y
2 2
1 1
2
1
2 2 2 3/ 2
2
2 12 3/ 2
cos( )
secsin sin tan
(tan 1)
x x
y x x
dx yE k k dx
r x y
k kd x y
y y
At a point P off axis:
Lecture 3-8
Continuous Charge Distribution 2: Charged Ring
Use symmetry!
2
3(
)
( )
(
/ )
/
kQ
kQ x x a
a x x a
2
Q
R
At point P on axis of ring:
ds
2 2
2 2 3/ 2
2 2 3/ 2
cos
( )
( )
x
dsE E k
x ax
k dsx a
xk Q
x a
Lecture 3-9Continuous Charge Distribution 3: Charged Disk
2 2 3/ 2
2 2 3/ 20
2 2 3/ 20
2 2 1 2
2
0
/
0
( )
2( )
2( )
2 ( )
12 1 0
1 ( / )
/ 2
x x
R
R
R
R whole plane and
xE E dE k dq
x a
xk ada
x a
akx da
x a
E
kx x a
k xR x
<= Independent of x
Superposition of rings!
At a point P on axis:Use the ring result
2 2 3/ 2( )x
xdE
aqk
xd
2dq a da
0
2
2 2
22
( )
k x R
Ek R kQ
x Rx x
Lecture 3-10
Continuous Charge Distribution 4: Charged Sheets
( ) ( ) E=const in each region Superposition!
Capacitor geometry
Lecture 3-11Physics 241 –Quiz 1A
The left half of a rod is uniformly charged with a positive charge density , whereas the right half is uniformly charged with a charge density of . What is the direction of the electric field at a point on the perpendicular bisector and above the rod as shown?
a) to the right b) to the leftc upd) downe) E is zeo.
Lecture 3-12Physics 241 –Quiz 1b
The upper half of a ring is uniformly charged with a positive charge density , whereas the lower half is uniformly charged with a charge density of . What is the direction of the electric field at a point on the perpendicular axis through the center of the ring and to the left of the ring as shown?
a) to the right b) to the leftc upd) downe) E is zeo.
Lecture 3-13Physics 241 –Quiz 1C
The upper half of a rod is uniformly charged with a positive charge density , whereas the lower half is uniformly charged with a charge density of . What is the direction of the electric field at a point on the perpendicular bisector and to the right of the rod as shown?
a) to the right b) to the leftc upd) downe) E is zeo.