Introduction,importance and scope of horticulture.pptx
228602.ppt
1. The divergence of E
If the charge fills a volume, with charge per unit
volume .
'
dq d
Where d is an element of
volume.
For a volume charge:
'
'
2
0
1 ( )
ˆ
( )
4 v
r
E R rd
r
R
2. Thus:
3 ' ' '
0
1
. 4 ( ) ( )
4 v
E r r r d
0
1
. ( )
E r
Gauss’s law in
differential form.
3
2
ˆ
. 4 ( )
r
r
r
' '
2
0
ˆ
1
. . ( )
4 v
r
E r d
r
3. Spherical polar coordinates (r, , )
r: the distance from the origin
: the angle down from the z axis is called polar angle
: angle around from the x axis is called the azimuthal
angle
sin cos
x r
sin sin
y r
cos
z r
ˆ ˆ ˆ ˆ
sin cos sin sin cos
r x y z
ˆ ˆ ˆ ˆ
cos cos cos sin sin
x y z
ˆ ˆ ˆ
sin cos
x y
ˆ ˆ
ˆ sin
dl drr rd r d
4. The Curl of E
For a point charge situated at origin:
2
0
1
ˆ
4
q
E r
r
Line integral of the field from some point a to some
other point b:
In spherical polar coordinates,
ˆ ˆ
ˆ sin
dl drr rd r d
2
0
1
.
4
q
E dl dr
r
5. 2
0
1
.
4
b b
a a
q
E dl dr
r
0
1
4 a b
q q
r r
True for electrostatic
field.
. 0
E dl
Apply stokes theorem:
0
E
The integral around a closed path:
6. Electric Potential
Basic concept:
The absence of closed lines is the property of vector
field whose curl is zero.
E is such a vector whose curl is zero.
Using this special kind of it’s property we can reduce a
vector problem: using V, we can get E very easily.
Vector whose curl is zero, is equal to the gradient of
some scalar function
E=0 the line integral of E around any closed loop
is zero (due to Stokes' theorem).
E V
7. otherwise you could go out along path (i) and return
along path (ii) and
Because the line integral is independent of path, we
can define a function
O is some standard reference point.
. 0
E dl
Therefore the line integral of E from
point a to point b is the same for all
paths.
. 0
E dl
( ) .
r
o
V r E dl
is called electric
potential
8. The potential difference between two points a and b:
( ) ( ) . .
b a
o o
V b V a E dl E dl
. .
b o
o a
E dl E dl
.
b
a
E dl
Using fundamental theorem for gradients:
( ) ( ) ( ).
b
a
V b V a V dl
So ( ). .
b b
a a
V dl E dl
E V
Electric field is the gradient of a scalar potential.
9. Electric Potential at an arbitrary point
•Electric potential at a point is given as the work done
in moving the unit test charge (q0) from infinity (where
potential is taken as zero) to that point.
• Electric potential at any point P is
Note that Vp represents the potential difference dV
between the point P and a point at infinity.
S.I. unit J/C defined as a volt (V) and 1 V/m = 1 N/C
.
p
p
V E ds
0
p
W
V
q
10. Potential Difference in Uniform E field
• Electric field lines always point in the direction of
decreasing electric potential.
Example: Uniform field along –y
axis (E parallel to dl)
.
B B
B A
A A
V V V E dl Edl
B
A
V Edl Ed
• When the electric field E is directed downward,
point B is at a lower electric potential than point A. A
positive test charge that moves from point A to point
B loses electric potential energy.
11. Potential Diff. in Uniform E field
Charged particle moves from A to B in uniform E
field.
.
b
a
V E ds
12. Potential Diff. In Uniform E field
(Path independence)
Show that the potential difference between point A and
B by moving through path (1) and (2) are the same as
expected for a conservative force field.
By path (1), . cos
B
A
V E dl El
13. path (2)
= 0 since E is to dl
. cos
C
A
V E dl Eh El
. .
C B
A C
V E dl E dl
14. Equipotential Surfaces (Contours)
VC = VB ( same potential)
In fact, points along this
line has the same
potential. We have an
equipotential line.
. 0
B
C
V E dl
No work is done in moving a test charge between any two
points on an equipotential surface.
The equipotential surfaces of a uniform electric field
consist of a family of planes that are all perpendicular to the
field.
15. Equipotential Surface
Equipotential Surfaces (dashed blue lines) and electric field lines
(orange lines) for (a) a uniform electric field produced by infinite
sheet of charge, (b) a point charge, and (c) an electric dipole. In all
cases, the equipotential surfaces are perpendicular to the electric
field lines at every point.
16. 16
Electrostatic Potential of a Point
Charge at the Origin
Q
P
r
'
2
'
0
2
'
0 0
4
4 4
r r
r
Q
V r E dl dr
r
Q dr Q
r
r
18. 18
Electrostatic Potential Resulting from
Continuous Charge Distributions
0
0
0
1
4
1
4
1
4
L
S
V
dl
V r
R
ds
V r
R
dv
V r
R
line charge
surface charge
volume charge
19. 19
Charge Dipole
• An electric charge dipole consists of a pair of equal
and opposite point charges separated by a small
distance (i.e., much smaller than the distance at
which we observe the resulting field).
d
+Q -Q
20. Dipole Moment
• Dipole moment p is a measure of the strength of the
dipole and has its direction.
p Qd
+Q
-Q
d
p is in the direction from the
negative point charge to the
positive point charge
23. • first order approximation from geometry:
cos
2
cos
2
d
r
R
d
r
R
d/2
d/2
lines approximately
parallel
R
R
r
24. 24
• Taylor series approximation:
cos
2
1
1
1
cos
2
1
1
cos
2
1
1
cos
2
1
1
1
r
d
r
R
r
d
r
r
d
r
d
r
R
1
,
1
1
:
Recall
x
nx
x
n
2
0
0
4
cos
2
cos
1
2
cos
1
4
,
r
Qd
r
d
r
d
r
Q
r
V
25. 25
• In terms of the dipole moment:
2
0
ˆ
4
1
r
a
p
V
r
26. Electric Potential Energy
of a System of Point Charges
1
0
1
4
q
V
r
2 ( )
W q V r
1 3 2 3
1 2
12 13 23
0 12 13 23
1
( )
4
q q q q
q q
W W W W
r r r
2
W F r q E r
q1
q2
2
b b
a a
W F dl q E dl
2[ ( ) ( )]
W q V b V a
2[ ( ) ( )]
W q V r V
and we know
27. The Energy of a Continuous Charge Distribution
For a volume charge density p,
1
2
W Vd
0 .E
Using Gauss’s Law:
0
( . )
2
W E Vd
So:
0
.( ) .
2
W E V d VE da
By doing integration by part:
and so,
V E
2
0
.
2 v s
W E d VE da
If we take integral over all space:
2
0
2 allspace
W E d
28. Poisson’s and Laplace’s Equation
E V
The fundamental equations for E:
0
. ;
E
0
E
2
. .( )
E V V
Gauss’s law then says that:
2
0
V
This is known as
Poisson’s equation.
In regions where there is no charge: 0
Poisson’s equation reduces to Laplace’s equation.
2
0
V
This is known as
Laplace’s equation.
