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Ch0208 - Chapter 8 Capacitors Dielectrics and Energy in...

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Chapter 8 Capacitors, Dielectrics, and Energy in Capacitors 53 8 Capacitors, Dielectrics, and Energy in Capacitors Capacitance is a characteristic of a conducting object. Capacitance is also a characteristic of a pair of conducting objects. Let’s start with the capacitance of a single conducting object, isolated from its surroundings. Assume the object to be neutral. Now put some positive charge on the object. The electric potential of the object is no longer zero. Put some more charge on the object and the object will have a higher value of electric potential. What’s interesting is, no matter how much, or how little charge you put on the object, the ratio of the amount of charge q on the object to the resulting electric potential ϕ of the object has one and the same value. ϕ q has the same value for any value of q . You double the charge, and, the electric potential doubles. You reduce the amount of charge to one tenth of what it was, and, the electric potential becomes one tenth of what it was. The actual value of the unchanging ratio is called the capacitance C sc of the object (where the subscript “sc” stands for “single conductor”). ϕ q C = sc (8-1) where: C sc is the capacitance of a single conductor, isolated (distant from) its surroundings, q is the charge on the conductor, and, ϕ is the electric potential of the conductor relative to the electric potential at infinity (the position defined for us to be our zero level of electric potential). The capacitance of a conducting object is a property that an object has even if it has no charge at all. It depends on the size and shape of the object. The more positive charge you need to add to an object to raise the potential of that object 1 volt, the greater the capacitance of the object. In fact, if you define q 1 to be the amount of charge you must add to a particular conducting object to increase the electric potential of that object by one volt, then the capacitance of the object is volt 1 1 q . The Capacitance of a Spherical Conductor Consider a sphere (either an empty spherical shell or a solid sphere) of radius R made out of a perfectly-conducting material. Suppose that the sphere has a positive charge q and that it is isolated from its surroundings. We have already covered the fact that the electric field of the charged sphere, from an infinite distance away, all the way to the surface of the sphere, is
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Chapter 8 Capacitors, Dielectrics, and Energy in Capacitors 54 indistinguishable from the electric field due to a point charge q at the position of the center of the sphere; and; everywhere inside the surface of the sphere, the electric field is zero. Thus, outside the sphere, the electric potential must be identical to the electric potential due to a point charge at the center of the sphere (instead of the sphere). Working your way in from infinity, however, as you pass the surface of the sphere, the electric potential no longer changes. Whatever the value
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