6.4.2 Calculating Electric Potential

• 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

• The graph of potential V against distance r for a negative or positive charge is:

The electric potential around a positive charge decreases with distance and increases with distance around a negative charge

• Unlike the gravitational potential equation, the minus sign in the electric potential equation will be included in the charge
• The electric potential varies according to 1 / r
  • Note, this is different to electric field strength, which varies according to 1 / r²

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)

Q = 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 and calculate final answer