Chapter 23

Potential Energy

Gravitation-Electrical, what’s the difference? so

Potential i.e. Voltage Potential is actually potential energy per unit charge V=U/q1 Volts (V)= Joule/Coulomb or J/C Recall E-field units were N/C Now we have a new unit for electric field: V/m So now we see that the electric field is expression of how the voltage is changing over space

The coolest thing about voltage It is a SCALAR! But it can tell us about the E-field and Forces!

How to calculate V Point charge (I put the minus sign in q) Collection of point charges Distribution of charge

Potential Difference (i.e. the Voltmeter) Potential difference between point A and point B

Equipotential Surfaces kq/r1 q kq/r2 E-field Lines Electric field lines are always perpendicular to equipotential surfaces

Equipotentials and Conductors Recall that all the charge of an isolated conductor resides on the surface All excess charge placed on an isolated conductor will distribute itself so that the surface will come to same potential because: If there is more or less charge between two points on the surface, there will be current flow until equilibrium re-established Furthermore, the electric field lines must be perpendicular to the surface at every point Because if there were components tangential to the surface, there would be

E from V

So

For radial E-fields only

Going back to “Work” Recall: But the work done is independent of path Just like in gravity

Back to where we started What if: Well, from point a to point a seems facetious, what about 0 to 2p? Such an integral is called a closed path integral and is expressed

Curls again Recall Curl(v) Curl=0 Curl=+, out of page (Right Hand Rule) Curl=0

Stoke Theorem

So

Funky Shapes and the Method of Recursion Needless to say, Gauss’s Law works well for 3 shapes: Sphere Cylinder Big, Flat Surfaces But what if you have a funny shape?

Some Definitions Recall that What is this quantity?

Laplacian

Laplace’s Equation This equation applies to many physical systems Gravitation Heat flow EM Quantum Mechanics

Relax! A numerical method to solve Laplace’s equation Requires Method of Relaxation Method of Recursion Finite Element Method Big Brothers Discrete Ordinate (reactor physics) Finite Difference Time Domain (RF engineering) Requires Known boundary conditions The ability to break geometry into small regular shaped volume elements Computer with sufficient memory (PVM)

Averaging Based on nearest neighbor averaging 5 5 5 ? ?=(5+0+0+0)/4 =1.25 This is the estimate of the voltage in this square

2-d and 3-d Loop thru multiple times You save the new values in a temporary array The new values replace the old values and you loop again For 3-d objects, there are 6 surfaces

More Realism By a simple average, you are assuming perfect conduction For different materials, you can slow up the transfer of voltage, heat, etc by using Heat: different thermal conductivity values Electricity: different electrical conductivity values Fluids: different viscosities

Electric Field Lines 1.0 3.5 2.0 0.0 4 5