1. 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
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2. 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).
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3. Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Some applications
Resistance
Capacitors
Microstrip lines capacitance
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4. 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
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5. 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
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6. 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

7. 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

8. 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
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9. 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
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10. 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

11. 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
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12. Two cases: Capacitance
Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Two cases: Capacitance
Parallel plate
Coaxial
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13. Parallel plate Capacitor
Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Parallel plate Capacitor
Charge Q and –Q or
Plate area, S
Dielectric, e
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14. Effect of er on Capacitance

15. 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
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16. Capacitors connection
Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Capacitors connection
Series
Parallel
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17. 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.

18. 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)
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19. Find the capacitance of
Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
They are connected in parallel and series
Barium titanate
polymer
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