18.2.1 Electric Potential

• In order to move a positive charge closer to another positive charge, work must be done to overcome the force of repulsion between them
• Energy is therefore transferred to the charge that is being pushed upon
  • This means its potential energy increases

• If the positive charge is free to move, it will start to move away from the repelling charge
  • As a result, its potential energy decreases back to 0

• This is analogous to the gravitational potential energy of a mass increasing as it is being lift upwards and decreasing and it falls
• The electric potential at a point is defined as:

The work done per unit positive charge in bringing a small test charge from infinity to a defined point

• Electric potential is a scalar quantity
  • This means it doesn’t have a direction

• However, you will still see the electric potential with a positive or negative sign. This is because the electric potential is:
  • Positive when near an isolated positive charge
  • Negative when near an isolated negative charges
  • Zero at infinity

• Positive work is done by the mass from infinity to a point around a positive charge and negative work is done around a negative charge. This means:
  • When a test charge moves closer to a negative charge, its electric potential decreases
  • When a test charge moves closer to a positive charge, its electric potential increases

• To find the potential at a point caused by multiple charges, add up each potential separately

• The electric potential in the field due to a point charge is defined as:

• Where:
  • V = the electric potential (V)
  • Q = the point charge producing the potential (C)
  • ε₀ = permittivity of free space (F m^-1)
  • r = distance from the centre of the point charge (m)

• This equation shows that for a positive (+) charge:
  • As the distance from the charge r decreases, the potential V increases
  • This is because more work has to be done on a positive test charge to overcome the repulsive force

• For a negative (−) charge:
  • As the distance from the charge r decreases, the potential V decreases
  • This is because less work has to be done on a positive test charge since the attractive force will make it easier

• Unlike the gravitational potential equation, the minus sign in the electric potential equation will be included in the charge
• The electric potential changes according to an inverse square law with distance

The potential changes as an inverse law with distance near a charged sphere

• Note: this equation still applies to a conducting sphere. The charge on the sphere is treated as if it concentrated at a point in the sphere from the point charge approximation

Part (a)

Step 1: Write down the known quantities

Radius of the dome, r = 15 cm = 15 × 10^-2 m

Potential difference, V = 240 kV = 240 × 10³ V

Step 2: Write down the equation for the electric potential due to a point charge

Step 3: Rearrange for charge Q

Q = V4πε₀r

Step 4: Substitute in values

Q = (240 × 10³) × (4π × 8.85 × 10^-12) × (15 × 10^-2) = 4.0 × 10^-6 C = 4.0 μC

Part (b)

 Step 1: Write down the known quantities

Q = charge stored in the dome = 4.0 μC = 4.0 × 10^-6 C

r = radius of the dome + distance from the dome = 15 + 30 = 45 cm = 45 × 10^-2 m

Step 2: Write down the equation for electric potential due to a point charge

Step 3: Substitute in values