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I-6 Capacitance and Capacitors

I-6 Capacitance and Capacitors. Main Topics. An Example of Storing a Charge. Capacity x Voltage = Charge . Various Types of Capacitors . Capacitors in Series . Capacitors in Parallel. Storing a Charge I.

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I-6 Capacitance and Capacitors

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  1. I-6 Capacitance and Capacitors

  2. Main Topics • An Example of Storing a Charge. • Capacity x Voltage = Charge. • Various Types of Capacitors. • Capacitors in Series. • Capacitors in Parallel.

  3. Storing a Charge I • In the 18th century people were amazed by electricity mainly by big sparks. The entertainers faced a problem of storing as big charges as possible. First they needed larger and larger “containers” but after a better solution was found! • Let’s have a conductive sphere ra=1 m. • A quiz: Can we put any charge on it?

  4. Storing a Charge II • The answer is NO! • We are limited by the breakdown intensity. In dry air Em 3 106 V/m. • The maximum intensity depends on the properties of the surroundings of the conductor and the conductor itself (there would be some limit even in vacuum). • If the maximum intensity is reached the conductor will discharge.

  5. Storing a Charge III • Using the Gauss’ law: E=0 in the sphere and E=kQ/ra2 close to the surface. • From relations of the potential and the intensity =kQ/ra in and on the sphere. • Combining these we get: =raE for r>ra • The maximum voltage and charge for our sphere is:  = 3 106 V  Q = 3.3 10-4 C.

  6. Storing a Charge IV • Then someone (in Leyden) made a miracle! He inserted a smaller sphere rb into the big one and grounded it. Since the smaller sphere was originally on the potential of the bigger one, the positive charges were repelled to the ground until the charge –Q remained on the smaller sphere. • The result: potential and field decreased!

  7. Storing a Charge V • The potential from the outer sphere: a = kQ/ra for rra ; a = kQ/r for r>ra • The potential from the inner sphere: b = -kQ/rb for rrb ; b = -kQ/r for r>rb • From the superposition principle: (r) = a(r)+ b(r) • The potential is zero outside the system!

  8. Storing a Charge VI • The potential on the inner sphere is here also the voltage between the spheres: V = (rb) = kQ(1/ra – 1/rb) = kQ(rb – ra)/rbra • If rb>ra/2 it starts to be interesting. Let: rb = 0.99ra and Q = 3.3 10-4 C  V = 3 10-4 V • We can charge further up to Q  3.3 10-2 C! • We have obtained a capacitor (condenser).

  9. The Capacitance • The voltage between any two charged conductors is generally proportional their charge Q = C V • The constant of proportionality C is called the capacitance. It is the ability to store the charge. • Its unit is called Farad. 1 F = 1 C/V

  10. Various Types of Capacitors • It makes sense to produce a device meant to store charge – a capacitor. • They are used to store a charge and at the same time a potential energy. • Most widely used are parallel plate, cylindrical and spherical capacitors.

  11. Quiz: Two Parallel Planes • Two large parallel planes are d apart. One is charged with a charge density , the other with -. Let Eb be the intensity between and Eooutside of the planes. What is true? • A) Eb= 0, Eo=/0 • B) Eb= /0, Eo=0 • C) Eb= /0, Eo=/20

  12. Determination of Capacitance • Generally, we find the dependence of Q on V and capacitance is the coefficient of the proportionality between them. • In the case of parallel plates of area A, distance d apart, charged to +Q and -Q: • Gauss’ law: E = /0 = Q/0A • Also: E=V/d  Q = 0AV/d  C = 0A/d

  13. Charging a Capacitor • To charge a capacitor we • either connect the capacitor to a power source one plate to its plus the other plate to its minus pole. The first will be charged with the positivecharge the other with the negative charge. The voltage of the power source will be across the capacitor. • or we ground (temporarily) one plate and charge the other as in our example.

  14. Capacitors in Series • Let us have two capacitors C1 and C2 in series. We can replace them by a single capacitor Cs = C1C2/(C1 + C2) • The total charge is the same on both capacitors • Q = Q1 = Q2 • The total voltage is the sum of both voltages • V = V1 + V2 • 1/Cs = V/Q = V1/Q+ V2/Q = 1/C1 + 1/C2

  15. Capacitors in Parallel • Let us have two capacitors C1 and C2 in parallel. We can replace them by a single capacitor Cp = C1 + C2 • The total charge is distributed on both caps • Q = Q1 + Q2 • The total voltage is the same on both caps • V = V1 = V2 • Cp = Q/V = Q1/V+ Q2/V = C1 + C2

  16. Homework • 24 – 4, 5, 6, 11, 26  due this Wednesday!

  17. Things to Read • Chapter 24 – 1, 2, 3

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