Area Under a Potential–Charge Graph

When charging a capacitor, the power supply pushes electrons from the positive to the negative plate
- It therefore does work on the electrons, which increases their electric potential energy
At first, a small amount of charge is pushed from the positive to the negative plate, then gradually, this builds up
- Adding more electrons to the negative plate at first is relatively easy since there is little repulsion
As the charge of the negative plate increases i.e. becomes more negatively charged, the force of repulsion between the electrons on the plate and the new electrons being pushed onto it increases
This means a greater amount of work must be done to increase the charge on the negative plate or in other words:

The potential difference V across the capacitor increases as the amount of charge Q increases

Charging a Capacitor

Charge on capacitor plates, downloadable AS & A Level Physics revision notes

As the charge on the negative plate builds up, more work needs to be done to add more charge

The charge Q on the capacitor is directly proportional to its potential difference V
- The graph of charge against potential difference is therefore a straight-line graph through the origin

Charge Potential-Difference Graph

EPE energy on graph, downloadable AS & A Level Physics revision notes

The electric potential energy stored in the capacitor is the area under the potential-charge graph

The electric potential energy stored in the capacitor can be determined from the area under the potential-charge graph which is equal to the area of a right-angled triangle:

$A r e a o f a r i g h t - a n g l e d t r i a n g l e = \frac{1}{2} \times b a s e \times h e i g h t$

Worked example

The variation of the potential V of a charged isolated metal sphere with surface charge Q is shown on the graph below. WE Area under a Potential–Charge question graph, downloadable AS & A Level Physics revision notes Using the graph, determine the electric potential energy stored on the sphere when charged to a potential of 100 kV.

Answer:

Step 1: Determine the charge on the sphere at the potential of 100 kV

WE Area under a Potential–Charge solution graph, downloadable AS & A Level Physics revision notes

- From the graph, the charge on the sphere at 100 kV is 1.8 μC

Step 2: Calculate the electric potential energy stored

- The electric potential energy stored is the area under the graph at 100 kV
- The area is equal to a right-angled triangle, so, can be calculated with the equation:

$A r e a = \frac{1}{2} \times b a s e \times h e i g h t$

- Substituting in the values gives:

$A r e a = \frac{1}{2} \times 1.8 μ C \times 100 k V$

$E l e c t r i c P o t e n t i a l E n e r g y = \frac{1}{2} \times (1.8 \times 10^{- 6}) \times (100 \times 10^{3}) = 0.09 J$

Exam Tip

Remember to always check the units of the charge–potential difference graphs. The charges can often be in µC or the potential difference in kV! The units must be in C and V to get a work done in J.

Area Under a Potential–Charge Graph (CIE A Level Physics)

Revision Note