Page 388 - Physics Coursebook 2015 (A level)

376
Cambridge International A Level Physics
WORKED EXAMPLE
Q/mC
V/V
Area of strip ΔW/mJ
Sum of areasW/mJ
1.0
1.0
0.5
0.5
2.0
2.0
1.5
2.0
3.0
4.0
If we apply the same idea to the capacitor graph (Figure 24.7a), then the area under the graph is the shaded triangle, with an area of 12 base × height. Hence the work done in charging a capacitor to a particular p.d. is given by:
W = 12 Q V
Substituting Q = CV into this equation gives two further
equations:
W = 12 C V 2
and
W = 12 Q 2
The energy W stored is proportional to the square of the potential difference V (W ∝ V 2). It follows that doubling the charging voltage means that four times as much energy is stored.
1 A 2000 μF capacitor is charged to a p.d. of 10 V. Calculate the energy stored by the capacitor.
Step1 Writedownthequantitiesweknow: C=2000μF
V=10V
Step2 Writedowntheequationforenergystored and substitute values:
W = 12 C V 2
W = 12 ×2000×10−6 ×102 =0.10J
This is a small amount of energy – compare it with the energy stored by a rechargeable battery, typically of the order of 10 000 J. A charged capacitor will not keep an MP3 player running for any length of time.
The area under the graph has been divided into strips to make it easy to calculate the energy stored. The first strip (which is simply a triangle) shows the energy stored when the capacitor is charged up to 1.0 V. The energy stored is:
12 Q V = 12 × 1 . 0 m C × 1 . 0 V = 0 . 5 m J
a Calculate the capacitance C of the capacitor.
b Copy Table 24.2 and complete it by calculating the areas of successive strips, to show how W depends onV.
c Plot a graph of W against V. Describe the shape of this graph.
Table 24.2 Data for Question 6.
C
These three equations show the work done in charging
up the capacitor. This is equal to the energy stored by the capacitor, since this is the amount of energy released when the capacitor is discharged.
We can also see from the second formula (W = 12 CV 2) that the energy W that a capacitor stores depends on its capacitance C and the potential difference V to which it is charged.
QUESTIONS
5 State the quantity represented by the gradient of the straight line shown in Figure 24.7a.
6 The graph of Figure 24.8 shows how V depends on Q for a particular capacitor.
V / V4
3 2 1 0
01234 Q / mC
Figure 24.8 The energy stored by a capacitor is equal to the area under the graph of voltage against charge.