Electric Potential Gradient (HL) (HL IB Physics)

• When a charge moves through an electric field, work is done
• The work done in moving a charge q is given by:

• Where:
  • W = work done on or by the field (J)
  • q = magnitude of charge moving in the field (C)
  • ΔV = potential difference between two points (J C⁻¹)

Electrical 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 equal to:

• Where:
  
  • V_f = final electric potential (J C⁻¹)
  • V_i = initial electric potential (J C⁻¹)

• The potential difference due to a point charge can be written:

• Where
  • Q = magnitude of point charge producing the potential
  • k = Coulomb constant (N m² C^–2)
  • r_f = final distance from charge Q (m)
  • r_i = initial distance from charge Q (m)

• An electric field can be described in terms of the variation of electric potential at different points in the field
  • This is known as the potential gradient

• The potential gradient of an electric field is defined as:

The rate of change of electric potential with respect to displacement in the direction of the field

• A graph of potential V against distance r can be drawn for a positive or negative charge Q
• This is a graphical representation of the equation:

• The gradient of the V-r graph at any particular point is equal to the electric field strength E at that point
• This can be written mathematically as:

• Where:
  • E = electric field strength (V m⁻¹)
  • ΔV = potential difference between two points (V)
  • Δr = displacement in the direction of the field (m)

• The negative sign is included to indicate that the direction of the field strength E opposes the direction of increasing potential

Graph of electric potential against distance

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

• The key features of this graph are:
  • All values of potential are negative for a negative charge
  • All values of potential are positive for a positive charge
  • As r increases, V against r follows a 1/r relation for a positive charge and a -1/r relation for a negative charge
  • The gradient of the graph at any particular point is equal to the field strength E at that point
  • The curve is shallower than the corresponding E-r graph

Determining potential from a field-distance graph

• The potential difference due to a charge can also be determined from the area under a field-distance graph
• A graph of field strength E against distance r can be drawn for a positive or negative charge Q
• This is a graphical representation of the equation:

• The area under the E-r graph between two points is equal to the potential difference ΔV between those points

The electric field strength E has a 1/r² relationship

• The key features of this graph are:
  • All values of field strength are negative for a negative charge
  • All values of field strength are positive for a positive charge
  • As r increases, E against r follows a 1/r² relation (inverse square law)
  • The area under this graph is the change in electric potential ΔV
  • The curve is steeper than the corresponding V-r graph