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
Electric Potential Due to a Point Charge
- 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
- 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