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

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Presentation on theme: "The divergence of E If the charge fills a volume, with charge per unit volume. Where d is an element of volume. For a volume charge: R."— Presentation transcript:

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

2 Thus: Gausss 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 True for electrostatic field. Apply stokes theorem: 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 its 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 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. Therefore the line integral of E from point a to point b is the same for all paths. 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 (q 0 ) from infinity (where potential is taken as zero) to that point. Electric potential at any point P is Note that V p 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

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) 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.

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) V C = V B ( 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 16 Electrostatic Potential of a Point Charge at the Origin Q P

17 17 Electrostatic Potential Resulting from Multiple Point Charges Q1Q1 P(R, ) O Q2Q2

18 18 Electrostatic Potential Resulting from Continuous Charge Distributions 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. +Q -Q p is in the direction from the negative point charge to the positive point charge

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

22 22 d/2 P

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

24 24 Taylor series approximation:

25 25 In terms of the dipole moment:

26 Electric Potential Energy of a System of Point Charges q1q1 q2q2 q3q3 and we know

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

28 Poissons and Laplaces Equation The fundamental equations for E: Gausss law then says that: This is known as Poissons equation. In regions where there is no charge: Poissons equation reduces to Laplaces equation. This is known as Laplaces equation.

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