1

Introductory Physics Hunter College CAPACITANCE

Capacitors are devices that store electrical energy. The energy is created by the distribution of charges on the plates of a capacitor. Unlike batteries that store electrical energy chemically, capacitors store energy via an electric field. Historically, electrical energy was stored in clumsy looking devices called Leyden Jars. Interestingly Benjamin Franklin referred to a collection of these devices as a battery.

Modern capacitors look nothing like this . They can take all shapes and forms.

In essence they are designed with 2 parallel plates with a space between them. When the plates become charged, an electrical field is created between them. Frequently, the space is simply filled with air. More often the space is filled with a material called a dielectric.which is there to increase the storage ability of the capacitor.

The figure above shows two parallel plates connected to a battery with no dielectric between them. The capacitance of two parallel plates here is dependent only on the area of the plates as well as the distance between them.

C is the capacitance, in farads; A is the area of overlap of the two plates, in square

meters; ε0 is the electric constant (ε0 ≈ 8.854×10−12 F⋅m−1); and

d is the separation between the plates, in meters;

Note that the value of capacitance depends on the geometry only and not on voltage or current.

2

When the two plates are connected to the battery, charges build up immediately: positive on one plate and negative on the other. The negative plate of the capacitor accepts electrons from the negative side of the battery, while the positive plate is losing electrons to the positive terminal side of the battery. Once fully charged, the capacitor possesses the same potential difference as the battery.

When connected to a circuit, the charges on the plates flow to the circuit discharging the capacitor and the potential difference across the plates decays to zero. The process of charging and discharging can then be repeated.

For a small capacitor, the capacity is small. But large capacitors can hold quite a bit of charge.Even nature shows the capacitor at work in the form of lightning. One plate is the cloud, the other plate is the ground and the lightning is the charge releasing between these two "plates." Obviously, in a capacitor that large, you can hold a huge amount of charge!

This operation is fundamental to all electrical circuits. If an oscilloscope is connected to a capacitor it will show a particular pattern of voltage and current changes increase and decay: a process used in timing a multitude of electrical and electronic operations ranging from a flash camera to computer circuits.

Charging Discharging

Our lab will explore the basic operations of a capacitor

Operations

Go to this PhET simulation to get started

https://phet.colorado.edu/sims/html/capacitor-lab-basics/latest/capacitor-lab-basics_en.html

Select this window

Spend some time manipulating the controls. Check all the boxes to better observe changes as the physical properties of the capacitor are varied.

3The physical properties can be varied by clicking and moving the arrows as shown.

Click and drag the voltmeter on the right onto the screen. Place the red lead above and black lead below the capacitor.

Note the reset button on the lower right.

Adjust the arrows to minimize the area and maximize the distance between them.

Move the little yellow circles at the top so that the capacitor is disconnected from the battery.

Toggle the switch to connect and disconnect the capacitor from the battery.

Slide the lever on the battery to +1.5 volts and then connect the capacitor to the battery .

Observe the electric field between the plates.

How does the electric field change when the battery voltage is set at -1.5 volts?The electric field changes when the battery voltage is set at -1.5 volts by the direction of

field, but still has the same amount of arrows as when the battery voltage is set at +1.5 volts.

Disconnect the battery from the capacitor.

Vary distance and area of the capacitor.

How does capacitance change as you vary distance?As distance increases, the capacitance decreases. As distance decreases, the capacitance

increases.How does capacitance change as you vary area?

As area increases, the capacitance increases. As area decreases, the capacitance decreases.Now fill in the table below. . Make sure the leads of the voltmeter are above ( red) and below(black) the capacitor.

Record the values indicated that you obtain from the window in the upper left corner.

Vary the area and the distance and observe changes in all listed variables.

What parameters change and what don’t?

4Adjust the battery at 1.5 volts and then connect the capacitor to the battery.

Now adjust the values of Area and Distance of the capacitors indicated. Note that as you vary one property the other is kept constant at the values indicated.

Status Condition Capacitance pfarads

Top PlateChargepC

Stored EnergypJ

Volts V

Distance Constant at 2mm

Area: Maximum400mm2

1.68 pF 2.52 pF 1.89 pJ 1.5 V

Area: Minimum100mm2

0.84 pF 1.26 pF 0.95 pJ 1.5 V

Area constant at 100mm2

Minimum distance 2mm

0.37 pF 0.55 pF 0.42 pJ 1.5 V

Maximumdistance10mm

0.10 pF 0.15 pF 0.11 pJ 1.5 V

After completing the table focus on the Maximum to Minimum ratios. Complete the table below.

Item Condition Capacitance Charge Energy Volt

RatioMax/Min

Distance 0.27 pF^2 0.27 pF^2 0.26 pJ^2

1 V

RatioMax/Min

Area 2 pF^2 2 pF^2 1.99 pJ^2

1 V

How do Capacitance, Charge, Energy and Voltage vary as the ratio of Maximum to Minimum Area when Distance is constant.

The Capacitance, Charge, and Energy have a similar number of 0.26, while the Voltage has a voltage of 1 V, when distance is constant.

How do Capacitance, Charge, Energy and Voltage vary as the ratio of Maximum to Minimum Distance when Area is constant.

5

Discharging through a light bulb

Now go to the other window.

Arrange the circuit to look like this

In this next phase you will observe the timing property of a capacitor.

Set the battery lever to 1.5 volts and leave it there for all conditions.

Note that the switch can toggle between the battery and the light bulb.

Do a few practice runs of charging and the capacitor and then, by moving the toggle switch to the other contact point , discharging it through the bulb.

Observe the varying intensity (brightness) and duration of the illumination of the light bulb for different distance and area of the capacitor. Describe the duration of illumination depend on the plate area and distance of the capacitor.

Now collect some qualitative data.

Set your smart phone or cell phone to work as a timer. You will be recording the time it takes for a capacitor to discharge under several conditions. If you do not have immediate access to a timer then do counts of “One Mississippi,2 Two Mississippi, etc.

Sketch (qualitatively) the brightness decay as a function of time for capacitor areas for the following conditions

1. Distance = 2mm , Areas = 100 mm2 and 400 mm2 on one graph. Label the approximate scale in time. Take a photos and paste the photo here.

2. Area =400 mm2, Distance = 2mm and 10 mm on one graph. Label the approximate scale in time. Take a photo and paste the photo here,

6

Record the values of capacitance, initial charge and initial energy in the table below under the 4 conditions of Area and Distance

Condition Areamm2

Distancemm

Capacitancepf

Charge pC

EnergypJ

Timeseconds

1 100 10

2 100 2

3 400 10

4 400 2

Note that the initial potential of the capacitor is 1.5 volts. This will also be the potential across the bulb, when it is turned on.

Ensure that the two leads of the voltmeter in contact with the two terminals of the bulb. Arrange things so that as soon as the bulb is connected to the capacitor you can start the timer.

Toggle the switch from the battery to the capacitor (left to right) . Record the amount of time it takes for the voltmeter to read zero. Do three trials of each arrangement so that you can get an average value. Make your averages to the nearest whole number.

Because this may be a little tricky to manipulate, just try to get time to the nearest second. Absolute precision is not crucial here. The purpose of this exercise is to vary the parameters and see how that affects the decay time.

Sketch two graphs from max to min edit

1. Distance ( y axis) versus time ( x axis)2. Area ( y axis) versus time ( x axis)

7

R-C CircuitsNow we will explore this timing process using another simulation. Let’s first watch this video

https://www.youtube.com/watch?v=YhKI-QTKFxE

When an empty (uncharged) capacitor is connected to a battery through a resistor, R, it does not charge up instantaneously. It takes time. The resistor in series with the capacitor functions to control the rate that charge Q is drawn to the plates of the capacitor. As the amount of charge increases on the plates so does the potential until it reaches that of the battery.

On the other hand , at the beginning, charge easily increases across the plates allowing current ( defined

as charge /time) to be at its maximum of VR . As it becomes harder for charges to accumulate on the

plates the current decreases towards zero.

The changes of the voltage across and current flowing toward the capacitor are not linear but follow the form of exponential rise and decay:

V C (t )=V 0(1−e¿¿ −tT

)I ( t )=−V 0

Re

−tT =−I 0 e

−tT ¿ Charging

V C (t )=V 0 e−tT I ( t )=

V 0

Re

−tT =I 0 e

−tT Discharging

where t represents the instantaneous time and T=RC is the time constant.

During the charging phase the curves for the voltage across the capacitor and the charging current look like the figure below.

8

Voltage changes get smaller towards the asymptote of maximum voltage while current changes slow down as the values dribble towards zero.

Discharging the capacitor reverses the process. The voltage across the plates drops rapidly, then slowly decreases towards zero. The current is initially high then decreases towards zero as well. But this time with an opposite sign.

9

(Source https://www.electronics-tutorials.ws/rc/rc_2.html)

For example, suppose a circuit with a 0.1-Farad capacitor and 10 -ohm resistor and 10-V battery , possessed an RC combination of 1 second. Then after two seconds of charging, starting from t=0 the voltage across the capacitor would be 6.32 volts ;after 2 seconds, it would have risen to 8.65 volts and so

on. The current would instantly rise to 10V

10 ohm=1 Amps, then decay so that at 1 second its value would

be 3.68 amps; at 2 seconds its value would decrease to 1.35 amps, and so on. The simple exponential decay says that after a long time the current would only approach zero and never reach zero.

From the curves shown above, the time it takes for the voltage and current to drop to a half of their initial values is

T 1/2=0.693T ≈ 0.7T

The halflife is easier to visualize on a chart. Later we will measure T through a measurement of T1/2.

We will study the charging and discharging processes by setting up a circuit, schematically shown below. We will display the voltage across the capacitor and the current flowing out of the capacitor as a function of time using oscilloscopes.

10

Charging:When switch S1 is closed and S2 open, the battery drives the charges to the capacitor. The charges will build up in the capacitor until it is fully charged to the same potential as the battery.

DischargingWhen S1 is open and S2 closed, the charges stored in the capacitor flow out of the capacitor, The process will continue until the charges stored in the capacitor are fully depleted.. Note that the branch with an open switch does not participate in the actions,

Visit the PhET site.https://phet.colorado.edu/en/simulation/legacy/circuit-construction-kit-ac

Click and drag the into the blue zone wire, resistor, capacitor , switch, stopwatch, and battery. Practice arranging and connecting them. Constructor the RC circuit following the schematic diagram shown above. When completed, the circuit should look like the figure below.

11Note that right clicking above a joint allows you to disconnect two elements.

You may watch the transient in ‘slow motion’ by repeatedly pressing the play /pause button located at the bottom of the screen,

Note where the leads of the two scopes are placed. The two leads for voltage above (red) and below ( black)the capacitor.

Adjust the capacitor values to 0.10 farads, the resistor to 20.0 ohms and the battery to 10 volts.

What is the maximum current? (V/R) What is the RC time constant?

Observation of charging and discharging processes

The capacitor can be charged by 1. pressing to pause,2. opening S2 and closing S1., 3. resetting the timer, and then4. pressing to play.

The capacitor can be discharged by 1. pressing to pause, 2. opening S1 , closing S2, 3. resetting the timer, and then4. pressing to play,

You may watch the “slow motion” of the processes by pressing and repeatedly while the process is in progress.

Note that if both S1 and S2 are closed, something bad will happen. When you see fire breaks out, open both switches and start all over again,

Observe the oscilloscope and watch the pattern of how the voltage/current values change over time.

Note the + and - signs above the scopes that change the scale of values.

Practice charging and discharging the capacitor while observing the time record. Practice the function as well as the +/- controls to adjust the scale of voltage and current readings.

Observe the patterns of voltage and current as the capacitor charges and discharges. You should have observed that when the capacitor is charging the current spikes at its maximum, then starts an exponential decay. Voltage starts from zero and climbs towards its asymptotic values after a period of time,

Adjust the current and voltage scales on the scopes so that the entire screens reflect their maximum values.

Measurement of RC time constant

The time constant T=RC is the time it takes for the voltage across the capacitor to reach 63.2% of its final value while charging, and 36.7% of its initial value while discharging. In the present setup, the voltage scale on the oscilloscope does not have fine graticules for accurate reading. We will measure the

12halflife, T1/2, which is the time it takes for the voltage to drop to 50% of its initial values. The time constant of the decay is related to the half-life by

T=T1 /2

0.7

To measure the half-life, initiate a charging process and measure the time it takes for the voltage to reach 50% of final value. Use the slow motion to help improve the accuracy. Repeat the process five times. Take the average of five measurements and calculate the time constant,

Measured Half-Life =

Average Half-Life =

Calculated RC time constant =

Theoretical Value of RC time constant =

Challenge question.

Describe a physical process in the natural world involving time that represents a parallel to an R-C timing process. ( eg radioactive decay, or water tank with a hole in the bottom). Explain,

Explain why in the real world it is not a good idea to close both S1 and S2 ?