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 electrostaticfield.
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 functionE=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) andBecause the line integral is independent of path, we can define a functionO is some standard reference point.is called electric potential
8 The potential difference between two points a and b: Using fundamental theorem for gradients:SoElectric 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 isS.I. unit J/C defined as a volt (V) and 1 V/m = 1 N/CNote 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),
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 Origin QP16
17 Electrostatic Potential Resulting from Multiple Point Charges Q2P(R,q,f)Q1O17
18 Electrostatic Potential Resulting from Continuous Charge Distributions line charge surface charge volume charge18
19 Charge DipoleAn 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-Q19
20 Dipole MomentDipole moment p is a measure of the strength of the dipole and has its direction.+Q-Qp is in the direction from the negative point charge to the positive point charge
21 Electrostatic Potential Due to Charge Dipole observationpointd/2+Q-QzqP21
26 Electric Potential Energy of a System of Point Charges q2q1and we knowq3
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.