AQA A Level Physics复习笔记7.4.7 Comparing Gravitational & Electrostatic Forces

• 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

   

• ΔV is 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

• 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 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) = 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