Measurement of Capacitance and Permittivity of Air

Object

Investigate the charging and discharging characteristics of a capacitor.

Determine the dielectric permittivity of air e_o.

Introduction

The capacitance of a pair of parallel plates of area A is

where d is the separation between the plates.

In this experiment a parallel plate capacitor is alternatively charged up (see figure 1(a)) to a voltage V_S and discharged through a resistor R_A (see figure 1(b)). The charge stored by the capacitor is small, so a high frequency switching between charge and discharge is used to give a detectable current (see figures 1(a) and 1(b)).

The charge (Q) stored in the capacitor at voltage V_S is given by:

Q = CV_S

The charge Q is discharged f times per second (f is the switching frequency of the electronic switching box). Hence the current I is given by:

I = CV_Sf

and substituting then:

I V_S f

Therefore:

• the current I is proportional to the supply voltage V_S.
• the current I is proportional to the frequency f.
• the current I is inversely proportional to the plate separation d.

The ammeter voltage V_A across the resistor R_A is related to the current by Ohm's Law. Hence:

V_A = I R_A

so that: V_A V_S f R_A

Apparatus

Combined Electronic Switching Circuit and Variable DC Power Supply, two aluminium plates, 100kW ammeter resistor (R_A), two digital voltmeters, metre rule, various 4mm test leads, plastic spacers.

Setting up the equipment

1. Place plastic spacers at the corners of the plates to hold them apart, the spacing between the plates can be varied in fixed steps by stacking up the spacers or using the range of spacers provided. To prevent short circuits plastic spacers MUST always hold the plates apart Always switch off the power before moving the plates or spacers!
2. The frequency of the Electronic Switching Circuit is displayed on the unit and can be varied using the frequency control. The frequency display shows zero when the frequency is too high!
3. The voltage of the DC supply is measured by voltmeter V_S (see Fig. 1).
4. The discharge current is measured by measuring the voltage V_A across the ammeter resistance R_A.

Experiment

1. For a fixed supply voltage V_S and fixed frequency f, measure the ammeter voltage V_A for a range of plate separations d. Plot a graph of I against 1/d. Is this graph a straight line?
2. For a given plate spacing d and fixed frequency f, double the supply voltage V_S. How does the ammeter voltage V_A change? Does this agree with theory?
3. With V_S at about 10 volts, measure the discharge current for a range of frequencies. Plot a graph of V_A versus f.
4. The gradient of the graph of ammeter voltage V_A against f should be equal to V_SR_A. Hence calculate the Permittivity of air e_o. What is the absolute error in your value of e_o?
5. The speed of light is given by the formula where m_o = 4*p*10^-7mkgC^-2. What is the value of c using your result for e_o? What is the absolute error in your value of c?

d (10 -3m)	1/d (m-1)
0.292 ± 0.003	3420 ± 40
0.428 ± 0.003	2340 ± 20
0.528 ± 0.001	1894 ± 4
0.733 ± 0.001	1364 ± 2
1.033 ± 0.002	968 ± 2
1.486 ± 0.003	673 ± 1
1.980 ± 0.002	505.1 ± 0.5
4.014 ± 0.001	249.13 ± 0.06
5.82 ± 0.02	171.7 ± 0.7

The first seven spacer thicknesses (0.292 - 1.980 mm) in the table above are cut from plastic card/sheet usually available in modelling shops in nominal thicknesses of 10, 20, 30, ... 80 thousanths of an inch. The last two thicknesses (4.014 & 5.82 mm) were made from some plastic & perspex sheet we had.

©  Mark Davison, 1997,  give feedback or ask questions   about this experiment.