Capacitance and Dielectrics

1. Capacitance and Dielectrics AP Physics C Montwood High School R. Casao

2. Definitions • Voltage—potential difference between two points in space (or a circuit). • Capacitor—device to store energy as potential energy in an E field. • Capacitance—the charge on the plates of a capacitor divided by the potential difference of the plates: C = q/V • Farad—unit of capacitance, 1F = 1 C/V. This is a very large unit of capacitance, in practice we use F (10-6) or pF (10-12). • Electric circuit—a path through which charge can flow. • Battery—device maintaining a potential difference V between its terminals by means of an internal electrochemical reaction. • Terminals—points at which charge can enter or leave a battery.

3. A capacitor consists of two uncharged conductors connected across the terminals of a voltage source (battery). The battery will move charge from one conductor to another until the voltage produced by the charge buildup is equal to the battery voltage. Definition of Capacitance

4. The charge on one conductor will be charged to +Q and the other conductor will be charged to -Q. An electric field E is established between the charged conductors. A capacitor is a device whose purpose is to store electrical energy which can then be released in a controlled manner during a short period of time. Definition of Capacitance

5. When the switch is closed, the battery establishes an electric field in the wire that causes electrons to move from the left plate into the wire and into the right plate from the wire. As a result, a separation of charge exists on the plates, which represents an increase in electric potential energy of the system of the circuit. This energy in the system has been transformed from chemical energy in the battery.

6. Definition of Capacitance • Capacitors are used when a sudden release of energy is needed. • Capacitance is the ratio of the charge Q on one conductor of the capacitor to the potential difference between the conductors, Vab. • Capacitance is always a positive quantity. • The unit for capacitance is the Farad (F). 1 F = 1 C/V; 1 F = 1 x 10-6 F.

7. The capacitance of a device depends on the geometric arrangement of the conductors. For a spherical conductor of radius R and charge +q (the conductor with the –q charge can be considered as a hollow conducting sphere of infinite radius): The voltage of the sphere is Capacitance of a Lone Spherical Conductor

8. Capacitance of a Spherical Conductor • The capacitance is given by: • eo is the permittivity of the insulating material (called the dielectric) between the plates.

9. The charge on each plate is distributed uniformly over an area A; Use Gauss's Law to calculate the electric field E between the plates of the charged capacitor: Parallel Plate Capacitor A +Q d A -Q

10. Parallel Plate Capacitor • The potential difference between the plates is: • Substitute into the capacitance equation to determine the capacitance for a parallel plate capacitor:

11. Parallel Plate Capacitor • The capacitance of a parallel plate capacitor is directly proportional to the area of the plates and inversely proportional to the distance between the plates. • When air or a vacuum is the dielectric material between the plates: • When another material is present as the dielectric material between the plates, the equation includes a dielectric constant k:

12. A cylindrical capacitor consists of a cylindrical conductor of radius a and length l surrounded by a coaxial cylindrical shell of radius b. The electric field E is perpendicular to the axis of the cylinders and is confined to the region between the cylinders. Cylindrical Capacitors

13. Determine the voltage between the two cylinders: The electric field for a cylinder is given by: Cylindrical Capacitors

14. Cylindrical Capacitors

15. The voltage V is negative based on the direction of the integration. Capacitance is a positive quantity, so use the magnitude of the voltage. You can also reverse the direction of the integration to Va – Vb. Substitute into the capacitance equation: Cylindrical Capacitors

16. Cylindrical Capacitors • An example of this type of capacitor is a coaxial cable. • The coaxial cable consists of two cylindrical conductors of radii a and b separated by an insulator. The cable carries currents in opposite directions in the inner and outer conductors. • The geometry is useful for shielding the electrical signal from external influences.

17. Cylindrical Capacitor w/ Gauss’s Law L a b +Q on center conducting cylinder -Q on outer conducting cylinder

18. Cylindrical Capacitor w/ Gauss’s Law L a b L

19. The spheres have uniform charge density and the electric field is uniformly distributed between the inner sphere of radius a and the outer sphere of radius b. The electric field E between the two spheres is given by: Spherical Capacitors

20. Determine the voltage between the two spheres: Spherical Capacitors

21. The voltage V is negative based on the direction of the integration. Capacitance is a positive quantity, so use the magnitude of the voltage. You can also reverse the direction of the integration to Va – Vb. Substitute into the capacitance equation: Spherical Capacitors

22. What is a lightning discharge ? • Friction forces between the air molecules within a cloud result in positively charged molecules moving to the lower surface, and the negative charges moving to the upper surface . • The lower surface induces a high concentration of negative charges in the earth beneath. • The resultant electric field is very strong, containing large amounts of charge and energy. • If the electric field is greater than than breakdown strength of air, a lightning discharge occurs, in which air molecules are ripped apart, forming a conducting path between the cloud and the earth.