E d a q encl o first make a gaussian surface a

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! E ! d ! A ! " = q encl ! o First make a Gaussian surface (a cylinder of length L and radius r) which encloses the inner wire (+q). ! E 2 ! rL = + q " o
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Capacitors This means that E is: Since E is directed radially outward we find: ! E = + q 2 !" o rL Next, integrate E from the negative outer shell to the positive inner wire to find V (radially inward). V = ! q 2 !" o L 1 r b a " dr V = ! ! E " d ! s ! + # V = ! + q 2 !" o rL ˆ r " # $ % & ' ( dr ˆ r ( ) b a )
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Capacitors Integrating 1/r yields: The capacitance only depends on the radii a and b and the length L V = ! q 2 !" o L ln a ( ) ! ln b ( ) " # $ % V = q 2 !" o L ln b a ! " # $ % & Finally use the definition of capacitance to find C: C = q ! V C = q q 2 !" o L ln b a ! " # $ % & C = 2 !" o L ln b a ! " # $ % &
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Capacitors Capacitors are usually used in circuits. A circuit is a collection of objects usually containing a source of electrical energy (like a battery). This energy source is connected to elements (like capacitors) that convert the electrical energy to other forms. We usually create a circuit diagram to represent all of the elements at work in the real circuit.
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Circuits For example, let’s say that we had two capacitors connected in parallel to a battery. In the circuit diagram we would represent a capacitor with a parallel line symbol: || The short line representing the negative terminal and the long line representing the positive terminal. | Also, in the circuit diagram we would represent a battery with a short line and a long line: i
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Capacitors The previous real circuit can then be drawn as a circuit diagram as follows:
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