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# Capacitors

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The Capacitor

The circuit boards found in your computer, phone, calculator, and pretty much every other electronic device you own often look much like the one shown above. Many circuit boards have capacitors, including this one. Capacitors can be used to smooth out electrical impulses or to turn constant electric currents into a series of impulses.

### The Capacitor

#### Sharing Charge

We already know that insulators are materials that do not allow electrons to flow through them easily. When you place excess electrons on an insulator, the electrons remain where you put them and do not move around. Conversely, conductors are materials that allow electrons to flow thorugh them freely. Since electrons repel each other, excess electrons on a conductor move to positions as far apart as possible. The difference can be seen in the image below, where the electrons on the insulator remain clumped near each other, while the electrons on the conductor have spread out to cover the whole surface.

Consider a charged conductor and an uncharged conductor. When the charged conductor is touched to the uncharged conductor, as far as the electrons are concerned, it has become one large piece of conducting material.  The electrons on the charged object will run onto the uncharged object until the density of the charge is evenly distributed over the entire surface of both objects. If the objects are the same size, the charge will be shared equally throughout. This method is occasionally used to divide a charge by half.

The earth is also a conductor. Touching a charged object to the earth is called grounding . When you touch a conductor to the earth, you allow the earth to share the charge. Since the earth is billions of times bigger than the object, the earth takes nearly all of the charge. The charged object that was grounded now has zero charge.

It is very easy to ground an object. All that is necessary is to touch a conducting wire to both the object and the earth. Electrical devices that run the risk of picking up a large static charge are grounded, meaning they are connected to the earth via such a conducting wire. Virtually all household appliances, especially washers and dryers, are grounded in this way to eliminate static charge. Similarly, large trucks, especially gasoline tankers, are grounded via a large chain hanging off the back to prevent sparks when fuel is being unloaded.

Spheres, whether hollow or filled, will always have the excess charge on the surface. In hollow spheres, the only place for an electron to exist is on the surface. Similarly, in a solid conducting sphere all the excess charge sits on the surface. This conclusion is a result of Gauss's Law, which tells us that the symmetry of the sphere and the fact that the electric field within the sphere is 0 forces the charge to the outside.

#### Capacitors Store Charge

Pieter Van Musschenbroek, a Dutch physician, invented a device in 1746 that could store electric charge. Though he named the device a Leyden jar, similar devices today are called capacitors.  A typical capacitor consists of a pair of parallel plates of area A  separated by a small distance d . The space between the two plates is most often filled with an insulator and frequently the plates are rolled into the form of a cylinder.

If voltage is applied to a capacitor, it quickly becomes charged. One of the parallel plates acquires a negative charge and the other an equal amount of positive charge. For a given capacitor, the amount of charge, Q , acquired by each plate is proportional to the potential difference, V

$Q=CV$

Where Q  is the charge in coulombs, V  is the voltage in volts, and C  is the particular capacitor's constant of proportionality. C  is also called the capacitance of the capacitor. The capacitance is the voltage the capacitor can reach before it discharges, allowing the voltage across the capacitor to drop to zero and the current to cross the capacitor. The SI unit for capacitance, according to the equation above, will be coulombs/volt, and this unit has been given the name farad , F . Most capacitors have capacitances in the range of one picofarad $(10^{-12}F)$  to one microfarad $(10^{-6} \ F)$ .

Example Problem: A sphere has a potential difference between it and the earth of 60.0 V when charged with $3.0 \times 10^{-6} \ C$ . What is the capacitance?

Solution: $C=\frac{Q}{V}=\frac{3.0 \times 10^{-6} \ coulombs}{60.0 \ volts}=5.0 \times 10^{-8} \ farads$

#### Summary

• Since electrons repel each other, when excess electrons are placed on a conductor, they will move to positions as far away from each other as possible.
• When a charged conductor is touched to an uncharged conductor, the electrons will migrate until the density of the charge becomes evenly distributed over the entire surface.
• Touching a charged object to the earth is called grounding.
• A charged conducting sphere will always have all the excess charge on its surface.
• A typical capacitor consists of a pair of parallel plates, separated by a small distance.
• $Q = CV$ , where Q  is the charge in coulombs, V  is the voltage in volts, and C  is the constant of proportionality, or capacitance.

#### Practice

The following video covers capacitors. Use this resource to answer the three questions that follow.

1. What do capacitors do?
2. What are the units of capacitance?
3. What is the formula for capacitance?

1. We have a large charged hollow sphere with a small hole in one side. The charge on the sphere is 1.00 C.  We insert another conducting sphere through the hole on an insulating stick and touch the inside of the charged hollow sphere and bring the second sphere outside the large sphere. What will be the charge on the second sphere?
2. Why does a charged object lose its charge when it is touched to the ground?
3. If a charged rubber rod is placed on a wooden table, the rubber rod will keep its charge for a long time. Why doesn’t the rod lose its charge immediately?
4. Both a  $3.3 \times 10^{-6} \ F$ and a  $6.8 \times 10^{-6} \ F$ capacitor are connected across a 15.0 V potential difference. Which capacitor has a greater charge and what is that charge?

#### Review

1. The two plates of a capacitor hold  $+2.5 \times 10^{-3} \ C$ and  $-2.5 \times 10^{-3} \ C$ of charge when the potential difference is 950 V. What is the capacitance?
2. The potential difference between two parallel wires in air is 120. V. They hold equal and opposite charges of $9.5 \times 10^{-11} \ C$ . What is the capacitance of the two wires?
3. How much charge flows from a 12.0 V battery when it is connected to a 9.00 microfarad capacitor?

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