Capacitance and Resistance Sandra Cruz-Pol, Ph. D. INEL 4151 ch6 Electromagnetics I ECE UPRM Mayagüez, PR

Resistance and Capacitance

To find E, we will use: Poisson’s equation: Poisson’s equation: Laplace’s equation: Laplace’s equation: (if charge-free) (if charge-free) They can be derived from Gauss’s Law

Resistance If the cross section of a conductor is not uniform we need to integrate: If the cross section of a conductor is not uniform we need to integrate: Solve for V Solve for V Then find E from its differential Then find E from its differential And substitute in the above equation And substitute in the above equation

P.E. 6.8 find Resistance of disk of radius b and central hole of radius a. a t b

Resistence Las resistencias reducen o resisten el paso de electrones 0Negro 1Marrón 2Rojo 3Naranja 4Amarillo 5Verde 6Azul 7Violeta 8Gris 9Blanco

Capacitance Is defined as the ratio of the charge on one of the plates to the potential difference between the plates: 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 Q and find V (Gauss or Coulomb) Assume V and find Q (Laplace) Assume V and find Q (Laplace) And substitute E in the equation. And substitute E in the equation.

Capacitance 1. Parallel plate 2. Coaxial 3. Spherical

Parallel plate Capacitor Charge Q and –Q Charge Q and –Q or or Dielectric,  Plate area, S

Coaxial Capacitor Charge +Q & -Q Charge +Q & -Q Dielectric,  Plate area, S c

Spherical Capacitor Charge +Q & -Q Charge +Q & -Q

What is the Earth's charge? The electrical resistivity of the atmosphere decreases with height to an altitude of about 48 kilometres (km), where the resistivity becomes more-or-less constant. This region is known as the electrosphere. There is about a volt (V) potential difference between the Earth's surface and the electrosphere, which gives an average electric field strength of about 6 V/metre (m) throughout the atmosphere. Near the surface, the fine-weather electric field strength is about 100 V/m. The electrical resistivity of the atmosphere decreases with height to an altitude of about 48 kilometres (km), where the resistivity becomes more-or-less constant. This region is known as the electrosphere. There is about a volt (V) potential difference between the Earth's surface and the electrosphere, which gives an average electric field strength of about 6 V/metre (m) throughout the atmosphere. Near the surface, the fine-weather electric field strength is about 100 V/m. The Earth is electrically charged and acts as a spherical capacitor. The Earth has a net negative charge of about a million coulombs, while an equal and positive charge resides in the atmosphere.

Capacitors connection Series Series Parallel Parallel

Resistance Recall that: Recall that: Multiplying, we obtain the Relaxation Time: Multiplying, we obtain the Relaxation Time: Solving for R, we obtain it in terms of C: Solving for R, we obtain it in terms of C:

So in summary we obtained : Capacitor C R=  C Parallel Plate Coaxial Spherical

P.E. 6.9 A coaxial cable contains an insulating material of  1 in its upper half and another material with  2 in its lower half. Radius of central wire is a and of the sheath is b. Find the leakage resistance of length L. A coaxial cable contains an insulating material of  1 in its upper half and another material with  2 in its lower half. Radius of central wire is a and of the sheath is b. Find the leakage resistance of length L. 11 22 They are connected in parallel because voltage across them is =

P.E. 6.10a Two concentric spherical capacitors with  1r =2.5 in its outer half and another material with  2r =3.5 in its inner half. The inner radius is a=1mm, b=3mm and c=2mm. Find their C. Two concentric spherical capacitors with  1r =2.5 in its outer half and another material with  2r =3.5 in its inner half. The inner radius is a=1mm, b=3mm and c=2mm. Find their C. 11 22 c We have two capacitors in series:

P.E. 6.10b Two spherical capacitors with  1r =2.5 in its upper half and another material with  2r =3.5 in its lower half. Inner radius is a=1mm and b=3mm. Find their C. Two spherical capacitors with  1r =2.5 in its upper half and another material with  2r =3.5 in its lower half. Inner radius is a=1mm and b=3mm. Find their C. 11 22 We have two capacitors in parallel: