# 7.5.1 Electric Potential

### 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
• Similarly, to move a positive charge away from a negative charge, work must be done to overcome the force of attraction 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 lifted upwards and decreasing as it falls
• The electric potential at a point is defined as:

The work done per unit positive charge in bringing a point 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 around an isolated positive charge
• Negative around an isolated negative charge
• 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 positive test charge moves closer to a negative charge, its electric potential decreases
• When a positive test charge moves closer to a positive charge, its electric potential increases
• To find the potential at a point caused by multiple charges, the total potential is the sum of the potential from each charge

The electric potential V decreases in the direction the test charge would naturally move in due to repulsion or attraction

#### Electric Potential Difference

• Two points at different distances from a charge will have different electric potentials
• This is because the electric potential increases with distance from a negative charge and decreases with distance from a positive charge
• Therefore, there will be an electric potential difference between the two points
• This is represented by the symbol ΔV
• Δis normally given as the equation

ΔV = Vf – Vi

• Where:
• Vf = final electric potential (J C-1)
• Vi = initial electric potential (J C-1)
• A difference in electric potential will give a difference in electric potential energy, which can also be calculated

### Electric Potential in Radial Field

• 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

#### Worked Example

A Van de Graaf generator has a spherical dome of radius 15 cm. It is charged up to a potential of 240 kV.

Calculate:

a) The charge is stored on the dome
b) The potential at a distance of 30 cm from the dome

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

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