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PHY 184. Spring 2007 Lecture 12. Title: Capacitor calculations. Announcements. 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.

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PHY 184

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## PHY 184

Spring 2007

Lecture 12

Title: Capacitor calculations

184 Lecture 12

### Announcements

• 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.

184 Lecture 12

### 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

184 Lecture 12

### Cylindrical Capacitor

• Consider a capacitor constructed of two collinear conducting cylinders of length L.

• The inner cylinder has radius r1 andthe outer cylinder has radius r2.

• Both cylinders have charge perunit length  with the inner cylinderhaving positive charge and the outercylinder 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.

184 Lecture 12

### 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 anexpression for the electric fieldbetween the two cylinders

184 Lecture 12

### Cylindrical Capacitor (3)

• As we did for the parallel plate capacitor, we define the voltage difference across the two cylinders to be V=V1 – V2.

• The capacitance of a cylindrical capacitor is

Note that C depends on geometrical factors.

184 Lecture 12

### Spherical Capacitor

• Consider a spherical capacitor formed by two concentric conducting spheres with radii r1 and r2

184 Lecture 12

### 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

184 Lecture 12

### Spherical Capacitor (3)

• To calculate the electric field, we use a Gaussian surfaceconsisting of a concentric sphere of radius r such that r1 < r < r2

• The electric field is always perpendicular to the Gaussian surface so

• … which reduces to

…makes sense!

184 Lecture 12

### 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

184 Lecture 12

### 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 r2 =  and r1 = R we find

…meaning V = q/4pe0R

184 Lecture 12

### Example

b=40 mm

a=38 mm

d

A ?

• 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.

Answers: (a) 84.5 pF; (b) 191 cm2

184 Lecture 12

### 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

184 Lecture 12

### Clicker Question

The capacitance is the constant of proportionality between change and voltage. It depends on the geometry not on the charge or voltage.

• 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

184 Lecture 12

### 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.

184 Lecture 12

### Capacitors in Parallel

.. key point for 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 …

184 Lecture 12

### Capacitors in Parallel (2)

• We can consider the three capacitors as one equivalent capacitor Ceq that holds a total charge q given by

• We can now define Ceq 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

184 Lecture 12

### Capacitors in Series

.. key point for capacitors in series

• Consider a circuit with three capacitors wired in series

• The positively charged plate of C1 is connected to thepositive terminal of the battery.

• The negatively charge plate of C1 is connected to thepositively charged plate of C2.

• The negatively charged plate of C2 is connected to thepositively charge plate of C3.

• The negatively charge plate of C3 is connected to thenegative 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 C1, which induces a negative charge on the opposite plate of C1, which induces a positive charge on C2, etc.

184 Lecture 12

### Capacitors in Series (2)

• Knowing that the charge is the same on all three capacitorswe can write

• We can express an equivalent capacitance Ceq 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.

184 Lecture 12

### Clicker Question

Ceq = 10 pF

• C1=C2=C3=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

184 Lecture 12

Ceq = 3 pF