Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Fields and Waves Lesson 3.3 ELECTROSTATICS - POTENTIALS Darryl Michael/GE CRD MAXWELL’S SECOND EQUATION Lesson 2.2 looked at Maxwell’s 1st equation: D D ds dv Today, we will use Maxwell’s 2nd equation: E 0 or E dl 0 Importance of this equation is that it allows the use of Voltage or Electric Potential POTENTIAL ENERGY Work done by a force is given by: dl F F dl If vectors are parallel, particle gains energy - Kinetic Energy If, F dl 0 Conservative Force Example : GRAVITY F • going DOWN increases KE, decreases PE • going UP increases PE, decreases KE POTENTIAL ENERGY If dealing with a conservative force, can use concept of POTENTIAL ENERGY For gravity, the potential energy has the form mgz Define the following integral: F dl Potential Energy Change P2 P1 POTENTIAL ENERGY Since E dl 0 and F q E We can define: Potential Energy = q E dl P2 P1 Also define: Voltage = Potential Energy/Charge V ( P2 ) V ( P1 ) E dl P2 P1 Voltage always needs reference or use voltage difference POTENTIAL ENERGY Example: Use case of point charge at origin and obtain potential everywhere from E-field Spherical Geometry Point charge at (0,0,0) E q 4 0 r 2 Integration Path r dl aˆ r infinity Reference: V=0 at infinity POTENTIAL ENERGY The integral for computing the potential of the point charge is: V (r ) V (r ) E dl r r 0 r V (r ) E dr r r r q 4 0 r dr 2 V (r ) q 4 0 r POTENTIAL ENERGY - problems Do Problem 1a Hint for 1a: R=a R=b R=r Use r=b as the reference Start here and move away or inside r<b region POTENTIAL ENERGY - problems For conservative fields: E dl 0 ,which implies that: E ds 0 , for any surface E 0 From vector calculus: f 0 ,for any field f Define: E V Can write: E f POTENTIAL SURFACES Potential is a SCALAR quantity Graphs are done as Surface Plots or Contour Plots Example - Parallel Plate Capacitor +V0/2 +V0 +V0 -V0 -V0 0 Potential Surfaces -V0/2 E-Field E-field from Potential Surfaces From: E V Gradient points in the direction of largest change Therefore, E-field lines are perpendicular (normal) to constant V surfaces (add E-lines to potential plot) Do problem 2 Numerical Simulation of Potential In previous lesson 2.2, problem 3 and today in problem 1, Given or Q E-field derive V derive Look for techniques so that , given or Q V derive Numerical Simulation of Potential For the case of a point charge: q q V V (r ) 4 0 r 4 0 r r Distance from charge r , is field point where we are measuring/calculating V r , is location of charge Numerical Simulation of Potential For smooth charge distribution: (r ) dv V (r ) 4 0 r r (r ) dl V (r ) 4 0 r r Volume charge distribution Line charge distribution Numerical Simulation of Potential Problem 3 (r ) dl V (r ) 4 0 r r Setup for Problem 3a and 3b Line charge: r origin r r r Location of measurement of V Line charge distribution Integrate along charge means dl is dz Numerical Simulation of Potential Problem 3 contd... Numerical Approximation Break line charge into 4 segments l r Charge for each segment r r q l l Segment length qi V 4 ch arg es 4 0 r ri Distance to charge Numerical Simulation of Potential Problem 3 contd... For Part e…. Get V(r = 0.1) and V(r = 0.11) Use: V V E V aˆ r aˆ r r r So..use 2 points to get V and r • V is a SCALAR field and easier to work with • In many cases, easiest way to get E-field is to first find V and then use, E V