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1/29/07184 Lecture 121 PHY 184 Spring 2007 Lecture 12 Title: Capacitor calculations

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1/29/07184 Lecture 122AnnouncementsAnnouncements Homework Set 3 is due tomorrow morning at 8:00 am. Midterm 1 will take place in class next week on Thursday, February 8. Practice exam will be posted in a few days. Second half of this Thursday’s lecture: review.

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1/29/07184 Lecture 123 Review of Capacitance The definition of capacitance is The unit of capacitance is the farad (F) The capacitance of a parallel plate capacitor is given by Variables: A is the area of each plate d is the distance between the plates

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1/29/07184 Lecture 124 Consider a capacitor constructed of two collinear conducting cylinders of length L. The inner cylinder has radius r 1 and the outer cylinder has radius r 2. Both cylinders have charge per unit length with the inner cylinder having positive charge and the outer cylinder having negative charge. We will assume an ideal cylindrical capacitor The electric field points radially from the inner cylinder to the outer cylinder. The electric field is zero outside the collinear cylinders. Cylindrical Capacitor

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1/29/07184 Lecture 125 Cylindrical Capacitor (2) We apply Gauss’ Law to get the electric field between the two cylinder using a Gaussian surface with radius r and length L as illustrated by the red lines … which we can rewrite to get an expression for the electric field between the two cylinders

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1/29/07184 Lecture 126 Cylindrical Capacitor (3) As we did for the parallel plate capacitor, we define the voltage difference across the two cylinders to be V=V 1 – V 2. The capacitance of a cylindrical capacitor is Note that C depends on geometrical factors.

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1/29/07184 Lecture 127 Spherical Capacitor Consider a spherical capacitor formed by two concentric conducting spheres with radii r 1 and r 2

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1/29/07184 Lecture 128 Spherical Capacitor (2) Let’s assume that the inner sphere has charge +q and the outer sphere has charge –q. The electric field is perpendicular to the surface of both spheres and points radially outward

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1/29/07184 Lecture 129 Spherical Capacitor (3) To calculate the electric field, we use a Gaussian surface consisting of a concentric sphere of radius r such that r 1 < r < r 2 The electric field is always perpendicular to the Gaussian surface so … which reduces to …makes sense!

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1/29/07184 Lecture 1210 Spherical Capacitor (4) To get the electric potential we follow a method similar to the one we used for the cylindrical capacitor and integrate from the negatively charged sphere to the positively charged sphere Using the definition of capacitance we find The capacitance of a spherical capacitor is then

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1/29/07184 Lecture 1211 Capacitance of an Isolated Sphere We obtain the capacitance of a single conducting sphere by taking our result for a spherical capacitor and moving the outer spherical conductor infinitely far away. Using our result for a spherical capacitor… …with r 2 = and r 1 = R we find …meaning V = q/4 0 R (we already knew that!)

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1/29/07184 Lecture 1212ExampleExample The “plates” of a spherical capacitor have radii 38 mm and 40 mm. a) Calculate the capacitance. b) Calculate the area A of a parallel-plate capacitor with the same plate separation and capacitance. b=40 mm a=38 mm d A ? Answers: (a) 84.5 pF; (b) 191 cm 2

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1/29/07184 Lecture 1213 Clicker Question Two metal objects have charges of 70pC and -70pC, resulting in a potential difference (voltage) of 20 V between them. What is the capacitance of the system? A) 140 pF B) 3.5 pF C) 7 pF D) 0

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1/29/07184 Lecture 1214 Clicker Question Two metal objects have charges of 70pC and -70pC, resulting in a potential difference (voltage) of 20 V between them. How does the capacitance C change if we double the charge on each object? A) C doubles B) C is cut in half C) C does not change The capacitance is the constant of proportionality between change and voltage. It depends on the geometry not on the charge or voltage.

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1/29/07184 Lecture 1215 Capacitors in Circuits A circuit is a set of electrical devices connected with conducting wires. Capacitors can be wired together in circuits in parallel or series Capacitors in circuits connected by wires such that the positively charged plates are connected together and the negatively charged plates are connected together, are connected in parallel. Capacitors wired together such that the positively charged plate of one capacitor is connected to the negatively charged plate of the next capacitor are connected in series.

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1/29/07184 Lecture 1216 Capacitors in Parallel Consider an electrical circuit with three capacitors wired in parallel Each of three capacitors has one plate connected to the positive terminal of a battery with voltage V and one plate connected to the negative terminal. The potential difference V across each capacitor is the same. We can write the charge on each capacitor as ….. key point for capacitors in parallel

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1/29/07184 Lecture 1217 Capacitors in Parallel (2) We can consider the three capacitors as one equivalent capacitor C eq that holds a total charge q given by We can now define C eq by A general result for n capacitors in parallel is If we can identify capacitors in a circuit that are wired in parallel, we can replace them with an equivalent capacitance

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1/29/07184 Lecture 1218 Capacitors in Series Consider a circuit with three capacitors wired in series The positively charged plate of C 1 is connected to the positive terminal of the battery. The negatively charge plate of C 1 is connected to the positively charged plate of C 2. The negatively charged plate of C 2 is connected to the positively charge plate of C 3. The negatively charge plate of C 3 is connected to the negative terminal of the battery. The battery produces an equal charge q on each capacitor because the battery induces a positive charge on the positive place of C 1, which induces a negative charge on the opposite plate of C 1, which induces a positive charge on C 2, etc... key point for capacitors in series

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1/29/07184 Lecture 1219 Capacitors in Series (2) Knowing that the charge is the same on all three capacitors we can write We can express an equivalent capacitance C eq as We can generalize to n capacitors in series If we can identify capacitors in a circuit that are wired in series, we can replace them with an equivalent capacitance.

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1/29/07184 Lecture 1220 Clicker Question C 1 =C 2 =C 3 =30 pF are placed in series. A battery supplies 9 V. What is the charge q on each capacitor? A) q=90 pC B) q=1 pC C) q=3 pC D) q=180 pC Answer: 90 pC C eq = 10 pF

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1/29/07184 Lecture 1221 Clicker Question C 1 =C 2 =C 3 =1 pF are placed in parallel. What is the voltage of the battery if the total charge of the capacitor arrangement, q1+q2+q3, is 90 pC? A) 180 V B) 10 V C) 9 V D) 30 V Answer: 30 volts Ceq = 3 pF

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