1. Capacitance and Resistance
Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Capacitance and Resistance
Sandra Cruz-Pol, Ph. D.
INEL 4151
Electromagnetics I
ECE UPRM
Mayagüez, PR
Electrical Engineering, UPRM (please print on BOTH sides of paper)

2. Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Resistance
If the cross section of a conductor is not uniform we need to integrate:
Solve Laplace eq. to find V
Then find E from its differential
And substitute in the above equation
Electrical Engineering, UPRM (please print on BOTH sides of paper)

3. 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.
Electrical Engineering, UPRM (please print on BOTH sides of paper)

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

5. Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Relaxation Time
Recall that:
Multiplying, we obtain the Relaxation Time:
Solving for R, we obtain it in terms of C:
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6. Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
P.E. 6.8 find Resistance of disk of radius b and central hole of radius a.
Laplace’s equation:
In cylindrical coordinates (if charge-free) =0 b a
Electrical Engineering, UPRM (please print on BOTH sides of paper)

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

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

10. 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)

11. Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Spherical Capacitor
Charge +Q & -Q
Electrical Engineering, UPRM (please print on BOTH sides of paper)

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

13. Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
In summary : C R
Parallel Plate
Coaxial
Spherical
Electrical Engineering, UPRM (please print on BOTH sides of paper)

14. Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Method of Images
 (or mirror charges) is a problem-solving tool in electrostatics.
Electrical Engineering, UPRM (please print on BOTH sides of paper)

15. Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Method of Images
 Use superposition
Valid only for top region (where Q is)
Electrical Engineering, UPRM (please print on BOTH sides of paper)