1. The divergence of E
If the charge fills a volume, with charge per unit volume . R
Where d is an element of
volume.
For a volume charge:

2. Thus:
Gauss’s law in differential form.

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

4. The Curl of E For a point charge situated at origin:
Line integral of the field from some point a to some other point b:
In spherical polar coordinates,

5. The integral around a closed path:
Apply stokes theorem:
True for electrostatic
field.

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).

7. Therefore the line integral of E from point a to point b is the same for all paths.
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.
is called electric potential

8. The potential difference between two points a and b:
Using fundamental theorem for gradients: So
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
S.I. unit J/C defined as a volt (V) and 1 V/m = 1 N/C
Note that Vp represents the potential difference dV between the point P and a point at infinity.

10. Potential Difference in Uniform E field
Example: Uniform field along –y axis (E parallel to dl)
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.
Electric field lines always point in the direction of decreasing electric potential.

11. Potential Diff. in Uniform E field
Charged particle moves from A to B in uniform E field. q

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),

13. path (2)
= 0 since E is  to dl

14. Equipotential Surfaces (Contours)
VC = VB ( same potential)
In fact, points along this line has the same potential. We have an equipotential line.
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. Electrostatic Potential of a Point Charge at the OriginQ P 16

17. Electrostatic Potential Resulting from Multiple Point ChargesQ2
P(R,q,f) Q1 O 17

18. Electrostatic Potential Resulting from Continuous Charge Distributions
 line charge
 surface charge
 volume charge 18

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 19

20. Dipole Moment
Dipole moment p is a measure of the strength of the dipole and has its direction. +Q -Q
p is in the direction from the negative point charge to the positive point charge

21. Electrostatic Potential Due to Charge Dipole
observation
point
d/2 +Q -Q z q P 21

22. P q
d/2
d/2 22

23. first order approximation from geometry:q
d/2
d/2
lines approximately
parallel

24. Taylor series approximation:24

25. In terms of the dipole moment:25

26. Electric Potential Energy of a System of Point Chargesq2 q1
and we know q3

27. The Energy of a Continuous Charge Distribution
For a volume charge density p,
Using Gauss’s Law:
So:
By doing integration by part:
and so,
If we take integral over all space:

28. Poisson’s and Laplace’s Equation
The fundamental equations for E:
Gauss’s law then says that:
This is known as Poisson’s equation.
In regions where there is no charge:
Poisson’s equation reduces to Laplace’s equation.
This is known as Laplace’s equation.