### Experiment 6

#### Charging and Discharging of a Capacitor

Objective:

The objective of this experiment is to verify the exponential behavior of capacitors during charging and discharging processes.

Equipment:

A capacitor, a resistance box, 2 multi-meters, connecting wires, a watch, a dc power source, and a 2-way switch

Theory:

A capacitor is a passive electric device that stores electric energy.  A parallel-plates capacitor is made of two parallel metallic surfaces, each of area A, separated by an insulation layer of thickness d, and it has a capacity of

where C is the capacity in Farads, A the area of each plate in m2,  d the insulation (dielectric) thickness in (m), and εo the permittivity of free space (vacuum) for electric field propagation expressed in F/m that reads Farads/meter.

The factor, κ , pronounced "kappa" denotes the dielectric constant, and depends on the material of the insulation layer.  The capacitance C does not depend on the material of the plates.  The dielectric constant εo is related to Coulomb's constant k by

Figure 1

In Fig. 1, at t = 0, the capacitor is uncharged.  As soon as the key in the circuit is closed, electrons flow from the negative pole of the battery toward the lower plate of the capacitor.  They distribute over the lower plate, making it negative.  At the same time, the repelled free electrons of the upper plate flow toward the positive pole of the battery.  This causes the upper plate to become positively charged.  This process does not happen suddenly.  It takes some time.  The current is greatest to begin with, and decreases as charges accumulate on the plates.   At the beginning the capacitor is empty; therefore, the voltage across it is zero, but as more and more charges build up on its plates, its voltage keeps increasing.  The voltage across the capacitor VC  asymptotically approaches the battery voltage VBat .

During the charging and discharging processes, the voltage across the capacitor and the current through it follow the following exponential equations:

(Charging)

Battery in Circuit

At t = 0,

VC = 0

IC = VB / R

As t → ∞

VC = VBat.

## IC = 0

(Discharging)

With battery removed, the initial capacitor voltage is  Vo= Qo/C making the initial current  Io = Vo/R

At t = 0,

VC = Qo /C

IC = Qo /(RC)

As t → ∞

VC = 0

## IC = 0

It is a good  idea to examine the values in the third and fourth columns by once setting t = 0 and once t   in the appropriate equations.  Note that the charge-voltage formula for a capacitor is Q = CV.  These exponential variations will be observed in this experiment.

Procedure:

Arrange a circuit as shown:

Figure 2

If a computer is used to directly graph VC and IC versus time via an electronic interface, there is no need for using large capacitance C and a large resistance R in order to have a large value for the time constant τ = RC If voltage, current, and time are measured by three different group members, then use of large capacitance and resistance is recommended in order to have a large value for τ so that relatively accurate measurements (readings) can be made.  The two-way switch  shown is first put in position 1 to start the charging process.  In this case, the group member who keeps track of time must also close the circuit at the same time he/she starts the stop watch.  He/she must also announce the time at equal intervals.  When he/she announces the time, two other group members read the current and voltage values.  A good value for τ is 20s, and intervals of 10s will give each experimenter enough time to read and record a value, and concentrate on the occurrence of the next value.  Obtaining 10 to 15 points for each of current and voltage is sufficient.  The data may be exchanged between the experimenters afterwards.  Do not disconnect the circuit.  This is because while preparation is underway for the discharging part of the experiment, the capacitor voltage keeps increasing asymptotically toward the battery voltage.

When all members are ready for the second part of the experiment (the discharging of the capacitor), the timekeeper must be ready to announce the starting time and at the same time put the two-way switch in position 2 as shown in Fig. 3 below:

Figure 3

The measured values may be recorded in tables as shown under the Data section.

Graphs:

Graph the following: For charging:  VC versus t    and   IC versus t, and

for discharging: VC versus t   and   IC versus t.

If ln(IC) is graphed versus t ,  a straight line will be obtained.  To understand why, let us consider the equation of IC (t) for the charging process:

Data:

Given:

VB   = the battery voltage (to be read at the start of charging)

VoC = the initial capacitor voltage (to be read at the start of discharging)

Let  R = 20kΩ and C = 1,000 μF such that τ = 20s.

Use other values if suggested by your instructor.

Measured:

Capacitor Charging:

Note:  IC  at t = 0 is Known.  IC = VB /R .

 t  (sec) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 VC (volts) 0 IC (Amps) VB/R

Capacitor Discharging:

Note: IC  at t = 0 is Known.  IC = VoC /R

 t    (sec) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 VC (volts) VoC IC (Amps) VoC/R

Calculations:

For the charging part:  Io = VB/R.

For the discharging part Io = VoC/R.

Comparison of the Results:

The accepted and measured values of τ may be used to obtain a percent error.  The other four graphs may be compared with the corresponding graphs in your physics book.

Conclusion: To be explained by students.

Discussion:  To be explained by students.