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

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Resistance and Capacitance

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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

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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

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

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

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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.

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Capacitance 1. Parallel plate 2. Coaxial 3. Spherical

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

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

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

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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 300 000 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 300 000 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.

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Capacitors connection Series Series Parallel Parallel

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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:

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

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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. 11 22 They are connected in parallel because voltage across them is =

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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. 11 22 c We have two capacitors in series:

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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. 11 22 We have two capacitors in parallel:

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III.A 3, Gauss’ Law.

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