Download 4. Capacitance

4. Capacitance
Background
The potential difference across a capacitor with capacitance C and stored charge Q is given by
V 
Q
C
If the capacitor is isolated from the voltage source after being charged up, and if the capacitance then changes,
the stored charge Q remains constant. As a result, the voltage V across it will change. An increase of
capacitance leads to a voltage drop, and a decrease of capacitance leads to a voltage rise.
The capacitance of a parallel plate capacitor in air with plates of one-sided area A that are separated by a
distance d from each other is given by
Cp  0
A
d
where  0  8.85 pF / m . The capacitance increases as d is reduced.
PART I Determination of the capacitance of an electrometer
Theory
An electrometer is essentially a voltmeter with very large internal resistance. Its internal circuit consists of a
large resistance in parallel with a capacitance C0 as shown
V
C0
Cp
When the leads from the electrometer are connected to those of a parallel plate capacitor with capacitance
C p (parallel connection), the combined capacitance is given by
C  C0  C p
1
If charge Q is introduced onto the positive plate of the parallel plate capacitance, it is shared with the internal
capacitor. As a result, the common voltage V across the capacitors obeys the relation
Q
 C p  C0
V
This voltage is indicated by the electrometer. The parallel plate capacitance C p can be varied by changing the
plate separation d . A plot of 1 V versus C p is expected to be a straight line, whose intercept with the horizontal
axis gives C0
Procedure
1. Connect the parallel plate capacitor with the electrometer using the coaxial cable as shown in the
diagram. The positive lead of the cable (red) is connected to the fixed plate.
Power supply
electrometer
cable
2. Make sure both the electrometer and the parallel plate capacitor are discharged. This can be
accomplished by pushing the zero button on the electrometer.
3. Set the scale of the electrometer at 100V.
4. Set the distance d between the plates at 1mm.
5. Set the voltage on the power supply to 40V and momentarily connect the positive terminal to the fixed
plate. A reading close to 40V should appear on the electrometer. Record it.
6. Set the plate separation d to 2,3,4 and 5mm, each time recording the electrometer reading.
7. Calculate C p for each d and plot 1 V against C p . Deduce the internal capacitance C0 from the graph.
8. Repeat step 4 through 7 with power supply voltage settings of 30V and 20V.
9. Calculate the average value of C0 .
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Part 2
Determination of Unknown Capacitance
When a unknown capacitor C x is charged to the potential difference V and then connected in parallel to an
uncharged known capacitor C , the potential difference will change to a new value V  . Because the total charge
does not change after the connection, we have
Q  C xV  C x  C V 
which can be solved to give C x 
V
C
V V 
Procedure:
1. Connect the leads of the electrometer to the unknown capacitor.
2. Bleed possible residual charge on the capacitor by temporarily connecting its terminals.
3. Set the power supply to 80V and charge up the capacitor. Then disconnect the power supply. Note
the electrometer reading and call it V .
4. Bleed any residual charge on the capacitor box, set the box capacitance to 0.1m F , and connect the
terminals to the unknown capacitor. Note the new reading on the electrometer and call it V  .
5. Calculate the unknown capacitance using the formula derived.
6. Repeat the procedures by setting the box capacitance to 0.2F .
7. Calculate the average value of C x and compare with its value listed on the capacitor.
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