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:
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
Thus: Gausss law in differential form.
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
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,
True for electrostatic field. Apply stokes theorem: The integral around a closed path:
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).
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
The potential difference between two points a and b: Using fundamental theorem for gradients: So Electric field is the gradient of a scalar potential.
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
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.
Potential Diff. in Uniform E field Charged particle moves from A to B in uniform E field.
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),
path (2) = 0 since E is to dl
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.
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 Origin Q P
17 Electrostatic Potential Resulting from Multiple Point Charges Q1Q1 P(R, ) O Q2Q2
18 Electrostatic Potential Resulting from Continuous Charge Distributions line charge surface charge volume charge
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
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 d/2 P
22 d/2 P
first order approximation from geometry: d/2 lines approximately parallel
24 Taylor series approximation:
25 In terms of the dipole moment:
Electric Potential Energy of a System of Point Charges q1q1 q2q2 q3q3 and we know
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:
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.