19.1.4 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 increase 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 ie. 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

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
• 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:

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

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

• • 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:

• • Substituting in the values gives: