Objects of the experiments
Determining the total capacitance of two capacitors in parallel connection and comparing with the capacitances of the
individual capacitors.
Determining the total capacitance of two capacitors in series connection and comparing with the capacitances of the
individual capacitors
Principles
Fig. 1 Parallel (above) and series connection (below) of capacitors
The capacitance C of a capacitor is the proportionality coefficient between the charge Q on the capacitor and the applied
voltage U:
Q = C ⋅ U (I).
When two capacitors with the capacitances C1 and C2 are
parallel-connected, they take the total charge
Q = Q1 + Q2 (II)
Q1, Q2: individual charges
because the voltage U is applied to both capacitors (see
Fig. 1). Because of Eq. (I), the capacitance of the parallel
connection is
C = C1 + C2 (III).
In a series connection, both capacitors take the same charge
Q. The applied voltage U is the sum of the individual voltages
U1 and U2:
U = U1 + U2 (IV).
The capacitances of the series connection therefore fulfil the
equation
1
C = 1
C1
+
1
C2
(V).
In the experiment, these relations are studied by means of two
plate capacitors with different capacitances C1 and C2.
The
capacitors are set up side by side, and both parallel or series
connection can be chosen. An insulating plate between the
two capacitors ensures that the charges on the capacitors
cannot influence each other through electrostatic induction.
Charges are measured with an electrometer amplifier operated
as a coulombmeter. Any voltmeter may be used to display the
output voltage UA.
From the reference capacitance CA
Q = CA ⋅ UA (VI).
is obtained. For example, at CA = 10 nF, UA = 1 V corresponds
to the charge Q = 10 nAs. In this case, the capacitance to be
measured is C =200 pF, if the voltage U = 50 V has been
applied.
Apparatus
Setup
Fig. 2 Experimental setup for measuring the capacitance of
parallel- and series-connected capacitors.
The experimental setup is illustrated in Fig. 2.
– Mount the pairs of large and small plates (as a distance
between the plates choose 6 mm for both pairs), and put
the polystyrene plate between them.
– Connect the voltmeter to the output of the power supply.
– Connect the positive pole of the power supply to socket B
of the two-way switch.
– Connect the negative pole of the power supply to the earth
socket of the electrometer amplifier.
– Connect the connection rod to the earth socket of the
electrometer amplifier with a connection lead.
– Connect socket C of the two-way switch to the input of the
electrometer amplifier.
– Plug the reference capacitor CA = 10 nF in at the electrometer amplifier.
– Supply the electrometer amplifier with voltage from the
plug-in unit.
– Connect the voltmeter to the output of the electrometer
amplifier.
The experiment
a) large plate capacitor
– Set the output voltage U of the power supply to 50 V.
– Connect the “inner” plate of the large plate capacitor to
socket A of the two-way switch and the “outer” plate to the
earth as shown in Fig. 3a.
– Establish the connection AC with the two-way switch, and
discharge the large plate capacitor with the connection rod.
– Hold the connection rod in your hand, and change the
two-way switch to the connection AB to charge the plate
capacitor.
– Set the two-way switch back to the connection AC,
measure the charge Q on the capacitor with the electrometer amplifier, and calculate the capacitance C from it.
b) small plate capacitor:
– Connect the small plate capacitor as shown in Fig. 3b.
– Establish the connection AC with the two-way switch, and
discharge the plate capacitor with the connection rod.
– Hold the connection rod in your hand, and charge the small
plate capacitor.
– Measure the charge Q, and calculate the capacitance C
from it.
c) parallel connection:
– As shown in Fig. 3c, connect the two inner plates to each
other and to socket A of the two-way switch; connect the
two outer plates to the earth.
– Discharge the parallel-connected capacitors, then charge
them, measure the charge Q, and calculate the capacitance
C.
d) series connection:
– Set the series connection up as shown in Fig. 3d.
– Discharge the series-connected capacitors, then charge
them, measure the charge Q, and calculate the capacitance
C
Measuring example
U = 50 V, d = 6 mm:
Evaluation
Parallel connection:
Application of Eq. (II) leads to C = 130 pF +240 pF = 370 pF.
Measuring result: C = 360 pF.
Series connection:
Application of Eq. (IV) leads to
1
C = 1
240 pF +
1
130 pF = 0.01186 1
pF.
From this C = 84.3 pF follows.
Measuring result: C = 88 p
Results
The capacitance of parallel-connected capacitors is equal to
the sum of the individual capacitances.
The reciprocal of the capacitance of series-connected capacitors is equal to the sum of the reciprocals of the individual
capacitances.
تعليقات
إرسال تعليق