Physics 212 Lecture 7, Slide 1 Physics 212 Lecture 7 Today's Concept: Conductors and Capacitance How are charges distributed on conductors? What is capacitance and how can we calculate it?

Main Point 1 First, charges in a conductor will always move so as to create a zero electric field at all points in the conductor. Consequently, all conductors are equipotentials and the electric field at the surface of a conductor is always perpendicular to that surface. Physics 212 Lecture 7, Slide 2

Main Point 2 Second, the capacitance C of a system composed of two spatially separated conductors (called a capacitor) is defined in terms of the potential difference V between the conductors that is produced when each conductor carries an equal amount Q of oppositely signed excess charge. In particular, C is defined to be equal to Q/V. Physics 212 Lecture 7, Slide 3

Main Point 3 Third, energy is stored in electric fields. The energy density (i.e., the energy per unit volume) present in an electric field E is proportional to the square of the electric field (i.e., u = ½ e 0 E 2 ). When a capacitor is charged, it has an energy U stored in the electric field between the conductors which is proportional to the product of the potential difference between the conductors and the separated charge (i.e., U = ½ QV) Physics 212 Lecture 7, Slide 4

Physics 212 Lecture 7, Slide 5 Checkpoint 1a 6 Two spherical conductors are separated by a large distance. They each carry the same positive charge Q. Conductor A has a larger radius than conductor B. Compare the potential at the surface of conductor A with the potential at the surface of conductor B. A. V A > V B B. V A = V B C. V A < V B

Physics 212 Lecture 7, Slide 6 Checkpoint 1b 7 The two conductors are now connected by a wire. How do the potentials at the conductor surfaces compare now? A. V A > V B B. V A = V B C. V A < V B

Physics 212 Lecture 7, Slide 7 Checkpoint 1c 8 What happens to the charge on conductor A after it is connected to conductor B by the wire? A. Q A increasesB. Q A decreases C. Q A doesn’t change

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Physics 212 Lecture 7, Slide 9 Checkpoint 2a A) Q 1 < Q o B) Q 1 = Q o C) Q 1 > Q o Two parallel plates of equal area carry equal and opposite charge Q 0. The potential difference between the two plates is measured to be V 0. An uncharged conducting plate (the green thing in the picture below) is slipped into the space between the plates without touching either one. The charge on the plates is adjusted to a new value Q 1 such that the potential difference between the plates remains the same.

Physics 212 Lecture 7, Slide 10 Checkpoint 2b A) C 1 > C o B) C 1 = C o C) C 1 < C o Two parallel plates of equal area carry equal and opposite charge Q 0. The potential difference between the two plates is measured to be V 0. An uncharged conducting plate (the green thing in the picture below) is slipped into the space between the plates without touching either one. The charge on the plates is adjusted to a new value Q 1 such that the potential difference between the plates remains the same. What happens to C 1 relative to C 0 ?

Physics 212 Lecture 7, Slide 11 Capacitance Capacitance is defined for any pair of spatially separated conductors 9 How do we understand this definition ??? +Q -Q d Consider two conductors, one with excess charge = +Q Consider two conductors, one with excess charge = +Q and the other with excess charge = -Q and the other with excess charge = -Q These charges create an electric field in the space between them These charges create an electric field in the space between them E This potential difference should be proportional to Q !! This potential difference should be proportional to Q !! The ratio of Q to the potential difference is the capacitance and only depends on the geometry of the conductors The ratio of Q to the potential difference is the capacitance and only depends on the geometry of the conductors V We can integrate the electric field between them to find the potential difference between the conductor We can integrate the electric field between them to find the potential difference between the conductor

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Physics 212 Lecture 7, Slide 13 Example Problem Two parallel plates of area A separated by a distance d carry equal an opposite charge Q 0. An uncharged conducting plate having thickness t is slipped midway between the plates. +Q 0 -Q 0 d t d +Q 1 -Q 1

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Physics 212 Lecture 7, Slide 15 BANG 31 Energy in Capacitors

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Physics 212 Lecture 7, Slide 17Calculation A capacitor is constructed from two conducting cylindrical shells of radii a 1, a 2, a 3, and a 4 and length L (L >> a i ). What is the capacitance C of this capacitor ? metal a1a1 a2a2 a3a3 a4a4 Conceptual Analysis: Strategic Analysis cross-section

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Physics 212 Lecture 7, Slide 19