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 Charge I In the end of 18 th century people were amazed by electricity, mainly by big discharges - sparks. The entertainers had noticed that different bodies charged to the same voltage contained different amounts of “electricity” (charge in our words) and produced sparks of different impact.

Storing Charge II So a problem had arisen to store as big charge as possible for the maximum voltage available. First they needed larger and larger “containers” but after a better solution was found! Let’s have a conductive sphere r i = 1 m. A quiz: If we were not limited by voltage, can we put any charge on it?

Storing Charge III The answer is NO! We are still limited by the breakdown intensity. In dry air E m  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 self-discharge. Rough surfaces would make things even worse.

Storing Charge IV Using the Gauss’ law: E = 0 within the sphere and E = kQ/r i 2 close to the surface. From relations of the potential and the intensity  = kQ/r i within and on the sphere. Combining these we get:  = r i E for r>r i The maximum voltage and charge for our sphere are :  = V  Q max = C.

Storing Charge V This voltage anyway far exceeded the limits of power sources at the time, which were, say, 10 5 V. On our sphere, such voltage would correspond to small charge : Q = Vr i /k = 10 5 / = C. This could originally be improved only by increasing of the sphere r i. Then someone (in Leyden, Germany) made a miracle! He inserted this sphere r i into a little bigger one r o, which he grounded. The sparks grew considerably!

Storing Charge VI The smaller sphere, charged with +Q, produced charge –Q on the inner surface of the bigger sphere and charge +Q in its outer surface. But when grounded the positive charge from the outer surface was repelled to the ground, so charge –Q remained on the outer sphere, particularly its inner surface. The result: potential of the charged sphere decreased, while the charge remained same!

Storing a Charge VII The potential from the inner sphere:  i = kQ/r i for r  r i ;  i = kQ/r for r > r i The potential from the outer sphere:  o = -kQ/r o for r  r o ;  o = -kQ/r for r > r o From the superposition principle:  (r) =  i (r)+  o (r) The potential is zero for r  r o !

Storing a Charge VI So the potential on the inner sphere is here also the voltage between the spheres: V i = kQ(1/r i – 1/r o ) = kQ(r o – r i )/r i r o Let r o = 1.01 m and V = 10 4 V  Q = C the charge increased 101 x! We have obtained a capacitor (condenser). (Q max would still be C, however !)

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

Various Types of Capacitors It makes sense to produce a device meant to store charge – a capacitor. The capacitance of capacitors should not depend on their surroundings. Capacitors are used to store a charge and at the same time a potential energy. Most widely used are parallel plate, cylindrical and spherical capacitors.

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

Determination of Capacitance I 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/  0 A Also: E=V/d  Q =  0 AV/d  C =  0 A/d

Determination of Capacitance II The potential on one sphere in the universe : V i = kQ/r i  C = r i /k The second “electrode” of this “capacitor” is the infinity or more probably ground, which is closer. The capacitance would be strongly influenced by presence of conductors in the near neighborhood.

Determination of Capacitance III In the case of spherical capacitor we had : V i = kQ(1/r i – 1/r o ) = kQ(r o – r i )/r i r o which corresponds to the capacitance : The capacitance doesn’t depend on near conductors, unless they are very close.

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 positive charge the other with the negative charge. The voltage of the power source will be across the capacitor, when equilibrium is reached. or we ground (temporarily) one plate and charge the other as in our example.

Capacitors in Series I Let us have two capacitors C 1 a C 2 in series. We ca replace them by a single capacitance If we charge one end while the other is grounded, both (all) capacitors will be charged by induction having the same charge : Q = Q 1 = Q 2 = …

Capacitors in Series II Connected electrodes must be at the same potential so the total voltage is the sum of voltages on both (all) capacitors : V = V 1 + V 2 

Capacitors in Parallel Let us have two capacitors C 1 a C 2 in parallel. We can replace them by one capacitance C p : C p = C 1 + C 2 Total charge is distributed between both (all) capacitors. Q = Q 1 + Q 2 Voltage on both (all) capacitors is the same V = V 1 = V 2  C p = Q/V = Q 1 /V+ Q 2 /V = C 1 + C 2

Absolute limit of charge Capacitance of a parallel plate capacitor (in vacuum) can be increased both by increasing the area of the plates and decreasing of their distance. Only the first way, however, leads to decrease of the electric field and thereby to increase the absolute limit of the the charge which can be stored! It would be actually better to ground the inner and charge the outer sphere of our spherical capacitor.

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

Things to Read This lecture covers : Chapter 24 – 1, 2, 3 Advance reading : Chapter 24 – 4, 5, 6