Resistance and Capacitance Electrostatic Boundary value problems; Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I Resistance and Capacitance Electrostatic Boundary value problems; Sandra Cruz-Pol, Ph. D. INEL 4151 ch6 Electromagnetics I ECE UPRM Mayagüez, PR https://www.youtube.com/watch?v=JEIkB_8v7qk Electrical Engineering, UPRM (please print on BOTH sides of paper)

Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I Last Chapters: we knew either V or charge distribution, to find E,D. NOW: we only know values of V or Q at some places (boundaries). Electrical Engineering, UPRM (please print on BOTH sides of paper)

Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I Some applications Resistance Capacitors Microstrip lines capacitance Electrical Engineering, UPRM (please print on BOTH sides of paper)

Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I To find E, we will use: Poisson’s equation: Laplace’s equation: (if charge-free) They can be derived from Gauss’s Law Electrical Engineering, UPRM (please print on BOTH sides of paper)

Depending on the geometry: Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I (It’s a scalar) We use appropriate coordinates: Cartesian: cylindrical: spherical: Solve for Laplace or Poisson depending on value of rv Use Separation of Variables to solve for V Apply Boundary values to find unique answer for V Electrical Engineering, UPRM (please print on BOTH sides of paper)

Resistance Defined as: The problem of finding the resistance of a conductor of nonuniform cross section can be solved by : Choose a suitable coordinate system Assume Vo a the potential difference between conductor’s terminals Solve Laplace’s eq. for V, then obtain E and I from and 4. Finally obtain R as Vo/I

Resistance PE 6.8 A disc of thickness t has radius b and a central hole of radius a. Take s = conductivity, find R between hole and rim of the disk Between the 2 fat sides of disk

Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I P.E. 6.8 Part a) find Resistance of disk of radius b and central hole of radius a. b a t Electrical Engineering, UPRM (please print on BOTH sides of paper)

Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I P.E. 6.8 Part b) find Resistance of disk Between the 2 fat sides of disk b a t Electrical Engineering, UPRM (please print on BOTH sides of paper)

Resist the flow of electrons Resistence Negro 1 Marrón 2 Rojo 3 Naranja 4 Amarillo 5 Verde 6 Azul 7 Violeta 8 Gris 9 Blanco Resist the flow of electrons

Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I Capacitance Is defined as the ratio of the charge on one of the plates to the potential difference between the plates: Assume Q and find V (Gauss or Coulomb) Assume V and find Q (Laplace) And substitute E in the equation. The (-) sign in V can be ignored because we want the absolute value of V Electrical Engineering, UPRM (please print on BOTH sides of paper)

Two cases: Capacitance Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I Two cases: Capacitance Parallel plate Coaxial Electrical Engineering, UPRM (please print on BOTH sides of paper)

Parallel plate Capacitor Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I Parallel plate Capacitor Charge Q and –Q or Plate area, S Dielectric, e Electrical Engineering, UPRM (please print on BOTH sides of paper)

Effect of er on Capacitance

Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I Dielectric, e Plate area, S + - c Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I Coaxial Capacitor Charge +Q & -Q Electrical Engineering, UPRM (please print on BOTH sides of paper)

Capacitors connection Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I Capacitors connection Series Parallel Electrical Engineering, UPRM (please print on BOTH sides of paper)

How to tell if C is in: Parallel: when they have same voltage across their plates. E is || to interface. Series: when E & D are normal to the dielectric interface.

So In summary we obtained: Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I So In summary we obtained: Capacitor C R (not derived) Parallel Plate Coaxial Spherical (not derived) Electrical Engineering, UPRM (please print on BOTH sides of paper)

Find the capacitance of Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I They are connected in parallel and series Barium titanate polymer Electrical Engineering, UPRM (please print on BOTH sides of paper)