Figure 1: capacitor circuit diagram with AC power source and resistor

Capacitance

IB Topic 11.3

Figure 2: conceptual diagram of capacitor with physical quantites

Capacitance IB 2018 1 DB & TELJR 6/24/2019 Parallel plate capacitor

The ability to store electric charge is described as its capacitance. A capacitor is a device that stores electric charge using charge separation. A Parallel-Plate Capacitor has equal and opposite charges stored on two identical flat parallel plates (as shown to the left).

The amount of charge the capacitor can hold is defined as its capacitance and is measured in the units of farads (F).

Experimental results show these factors that determine capacitance:

Factors: Relation: Relationship:

Permittivity is a measure of electric flux created per unit of charge. The greater the permittivity, the less electric flux exists per charge. In more simple terms, an object with a large permittivity will have a small electric field across it.

Permittivity is most relevant when dealing with dielectrics in capacitors. Dielectrics become polarized in the presence of an electric field. The greater the polarization, the greater the induced electric field, which counteracts the external electric field. The net electric field inside of an object with a permittivity not equal to the permittivity of free space is less than the electric field outside of the object. This allows more charge to be built up on the plates of the capacitor.

Electric permittivity is symbolized with the letter . The smallest possible permittivity is the permittivity of free space (a vacuum), which is symbolized as 0 and can be found in your DATABOOK; 0 =

So; we find that the charge, q, is proportional to electric permittivity.

푉퐴 푉퐴 푞 = ε Databook0 for air or a vacuum 푞 = ε 푑 0 푑 Capacitance IB 2018 2 DB & TELJR 6/24/2019 Capacitance

The capacitance, C, of a parallel plate capacitor is defined as,

Written concept Formula

C =

Physics data book

Capacitance, C, has the derived unit of , with symbol . Its fundamental units are: , and it is (large, “meh sized”, extremely small ) unit. So we see the prefixes , , usually.

Combining the above two formulas reveals the capacitance formula for a parallel plate capacitor expressed with electric permittivity.

C =

Physics data book

Practice problem 1. A parallel plate capacitor with square plates whose sides are 27 cm long and 3.5 mm apart is separated by air. What is its capacitance? PSYW

Capacitance IB 2018 3 DB & TELJR 6/24/2019

Electric Field Between Two Parallel Plates

Review the figures to the right and describe the electric field between the two plates of a parallel plate capacitor. Description:

Capacitance IB 2018 4 DB & TELJR 6/24/2019 Dielectric materials

Electric permittivity is symbolized with the letter . The smallest possible permittivity is the permittivity of free space (a vacuum), which is symbolized as 0 and can be found in your DATABOOK; 0 =

The ratio of a dielectric’s permittivity to the permittivity of free space is called the relative permittivity A.K.A. “dielectric constant", and it is symbolized with the Greek letter eplisilon , ε.

ε ε푟 = Databook0 ε표

Dielectric constant Definition: A dielectric is an insulator with molecules that can be polarized in electric fields. It is placed between the conductors of a capacitor.

Purposes of a Dielectric: It prevents the conductors from touching In practice, this allows capacitors to be built with very little distance separating the plates.

It reduces the likelihood that sparks will jump from one conductor to the other As you can see from the formula C = Q/V, capacitance is the ratio of the amount of charge that is stored on a capacitor to the voltage across that capacitor. So, in order to maintain a constant ratio, as the voltage increases, so does the charge on each plate. Eventually, there is so much charge on the two plates that something undesirable happens—a spark jumps between the plates. When this happens, the capacitor’s breakdown rating has been exceeded. Remember, the purpose of the capacitor is to store charge on the plates, so if charges are jumping between the plates, the capacitor is not functioning properly.

We’ve already seen examples of jumping charges, or “sparks,” with the Van De Graaff generator. Someone stood close to the Van De Graaff generator as it built up charge. While the charge was growing, the potential difference between it and the person increased. Eventually, that voltage was so high that a spark jumped the gap between them. But imagine instead if we had put a big wall of rubber between the person and the Van De Graaff generator. It would have been much harder for charges to jump through the wall of rubber than it was for them to jump through the air. In this case, the rubber wall is the dielectric.

It increases the capacitance of the capacitor Capacitance is an inherent characteristic of the capacitor. In other words, the only way to change the capacitance (of a simple circuit) is to change the capacitor – this is similar to how the only way to change the resistance (of a simple circuit) is to change the resistor, and the only way to change the voltage (of a simple circuit) is to change the voltage source. Adding a dielectric changes the capacitor. Capacitance IB 2018 5 DB & TELJR 6/24/2019

Use the space below to explain why the dielectric causes the capacitance to increase.

C =

Physics data booklet

Practical capacitors and their use:

1. 2. 3.

Capacitance IB 2018 6 DB & TELJR 6/24/2019 Test Your Understanding

1. Imagine you had a capacitor with no dielectric that was connected to a voltage source. The capacitor would build up charge until the potential difference of its plates equaled the potential difference of the voltage source. Now imagine you add a dielectric into the capacitor while it’s still connected to the voltage source.

a. What would happen to the capacitance of the capacitor?

b. What would happen to the voltage of the capacitor?

c. What would happen to the charge of the capacitor?

2. Again, imagine you had a capacitor with no dielectric that was connected to a voltage source. However, this time you disconnect the capacitor from the voltage source, and then insert the dielectric.

a. What would happen to the capacitance of the capacitor?

b. What would happen to the voltage of the capacitor?

c. What would happen to the charge of the capacitor?

Capacitance IB 2018 7 DB & TELJR 6/24/2019 Energy Stored in a Capacitor (Thinking it through)

Connecting a capacitor to a voltage source causes charge to move across the plates. This means that an E-field is created, and therefore Work must be done to move the next charge, as charge builds, E-field builds and subsquient work done in moving the charge increases. as more charge moves, A data plot of charge – voltage The gradient is the q – V graph represents the capacitance . The area under the under the q – V graph represents .

Energy transferred to a capacitor = average p.d. (0 – V) X charge on the capacitor when the p.d. is V

Derive the energy formula for a capacitor: The area is ½ b * h

E=

Physics data booklet

How much energy and charge is stored in a 5.0 μF capacitor when it is connected to a 120 V power source?

Capacitance IB 2018 8 DB & TELJR 6/24/2019 Energy Stored in a Capacitor (Answers)

The charge on the capacitor, Q, is proportional to the p.d., voltage, across it.

The gradient of this relationship is the capacitance of the capacitor.

훥푞 퐶 = => Δq = CΔV 훥푉

We know that Work = Energy = V * q for a charge at a p.d.

The figure shoes us that V and q are not constant, only Capacitance is. So if we take the average p.d. for the charge we get.

1 퐸 = 푉푞 2

Energy stored in a capacitor is equal to the area under the curve of the V-Q graph. Which also yields the above equation.

In trerm of capaticance, C, we can do a substitution from the definition and gradient above; q = CV in to the equation.

1 1 2 퐸 = 푉(퐶푉) = 퐶푉 2 2

Physics data booklet

Capacitance IB 2018 9 DB & TELJR 6/24/2019 Capacitors in Series and Parallel circuits

Ceq = 5 uF Ceq = 30 uF

When capacitors are added in series the total capacitance is given by this formula/relationship...

When capacitors are added in parallel the total capacitance is given by this formula/relationship...

Q.1. Three 100 pF capacitors are to be added in series and then in parallel. Determine the equivalent resistance of each. Series: Parallel:

If they are connected to a 100 Volt source, what will be the total charge and energy stored on each system of capacitors?

Capacitance IB 2018 10 DB & TELJR 6/24/2019 Q2. Calculate the pd across the 4µF capacitor shown in the top figure on the left. PSYW and explain your thoughts on the solution.

Q3 Calculate the pd across the 4µF capacitor shown in this bottom figure on the left. PSYW and explain your thoughts on the solution.

Capacitance IB 2018 11 DB & TELJR 6/24/2019 Charging a capacitor - Current

Closing the switch causes current to flow as the capacitor charges, this takes time. In the first moment, capacitance doesn’t play a part in current flow!

The relationship for this is ε = Vresistor = I R

AS The pd across the capacitor and resistor increases,.

ε = Vresistor + Vcapacitor

The result is a smaller voltage across the resistor, so the current in the circuit must be smaller.

As the charge across the capacitor increases, pd across the resistor drops to zero and the current in the circuit is zero and the capacitor is charged. Now; VC = ε and VR = 0.

풕 − 푰 = 푰풐풆 흉

Where; 흉 is “time constant” Really 흉 = 푹푪 so the units are seconds. When t = RC, the current will be about 1/3 of the original current.

Figure 3 NOTE: Vo is emf of source

Capacitance IB 2018 12 DB & TELJR 6/24/2019 Discharging a capacitor, C, through a resistor, R.

Time constant of an RC circuit, 흉 = RC

풕 − 풒 = 풒풐풆 흉

풕 − 푽 = 푽풐풆 흉

풕 − 푰 = 푰풐풆 흉

Figure 4 p.d. - time graph of charging and discharging RC - circuit.

As we can see from the data above, a capacitor discharges more slowly than is charges as a rule.

Capacitance IB 2018 13 DB & TELJR 6/24/2019 Resistors and Capacitors A student constructs the circuit shown below and makes measurement of current. The capacitor is uncharged

a) Label the + and – side of the battery. Show the direction of the current. b) Label the + and – side of the capacitor. Show the direction of the current produced by the capacitor. c) She records her data (as shown). Using her data, make a summary statement about her findings. Offer and explanation. 10 V

Time (s) Voltage of Current (A) Capacitor(V) 0 0 10 1 1 5 2 4 2.5 3 7 1.25 4 9 0.625 d) Sketch the graph of Current vs Time 5 10 0

After the capacitor is fully charged the switch is moved into its second position.

Capacitance IB 2018 14 DB & TELJR 6/24/2019 a) Will current flow in the circuit? If so what direction?

b) When will the current stop flowing?

c) How can the capacitor be recharged?

Q.1. A 12 V battery is connected in series to a 3 ohm resistor and an initially uncharged capacitor of 30 pF. Determine the current in the circuit. a) immediately after the switch is closed b) after a long time.

c) Determine the charge and energy stored on the capacitor.

Capacitance IB 2018 15 DB & TELJR 6/24/2019 Q.2. A resistance-capacitor circuit is connected as shown. The capacitor does not have any charge and has a rating of 50 nF. The value of the resistors is 50 ohms. The value of the battery is 30 volts.

a) Determine the current in the circuit just as the battery is connected to the circuit. b) Determine the current in the circuit after a long time. c) Determine the charge and energy stored on the capacitor.

Conclusions:

Capacitance IB 2018 16 DB & TELJR 6/24/2019