10.2.3 Potential Gradient & Difference

Electric Potential Gradient

• An electric field can be defined in terms of the variation of electric potential at different points in the field:

The electric field at a particular point is equal to the negative gradient of a potential-distance graph at that point

• The potential gradient is defined by the equipotential lines
  • These demonstrate the electric potential in an electric field and are always drawn perpendicular to the field lines

Equipotential lines around a radial field or uniform field are perpendicular to the electric field lines

• Equipotential lines are lines of equal electric potential
  • Around a radial field, the equipotential lines are represented by concentric circles around the charge with increasing radius
  • The equipotential lines become further away from each other
  • In a uniform electric field, the equipotential lines are equally spaced

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

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

• The electric field strength is equivalent to this, except with a negative sign:

• Where:
  • E = electric field strength (V m^-1)
  • ΔV = change in potential (V)
  • Δr = displacement in the direction of the field (m)

• The minus sign is important to obtain an attractive field around a negative charge and a repulsive field around a positive charge

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

• The electric potential changes according to the charge creating the potential as the distance r increases from the centre:
  • If the charge is positive, the potential decreases with distance
  • If the charge is negative, the potential increases with distance

• This is because the test charge is positive

Gravitational Potential Gradient

• A gravitational field can be defined in terms of the variation of gravitational potential at different points in the field:

The gravitational field at a particular point is equal to the negative gradient of a potential-distance graph at that point

• The potential gradient is defined by the equipotential lines
  • These demonstrate the gravitational potential in a gravitational field and are always drawn perpendicular to the field lines
• The potential gradient in a gravitational field is defined as:

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

• Gravitational field strength, g and the gravitational potential, V can be graphically represented against the distance from the centre of a planet, r 

• Where:
• g = gravitational field strength (N kg^-1)
• ΔV = change in gravitational potential (J kg^-1)
• Δr = distance from the centre of a point mass (m)

• The graph of V against r for a planet is:

• The key features of this graph are:
  • The values for V are all negative
  • As r increases, V against r follows a -1/r relation
  • The gradient of the graph at any particular point is the value of g at that point
  • The graph has a shallow increase as r increases

• To calculate g, draw a tangent to the graph at that point and calculate the gradient of the tangent
• This is a graphical representation of the equation:

where G and M are constant

• • Earth’s mass, M_E = 5.97 × 10²⁴ kg
  • Radius of the Earth, r_E = 6.38 × 10⁶ m
  • Initial distance, r₁ = 3 × r_E = 3 × (6.38 × 10⁶) m = 1.914 × 10⁷ m
  • Final distance, r₂ = 1 × r_E = 6.38 × 10⁶ m
  • Gravitational constant, G = 6.67 × 10⁻¹¹ m³ kg⁻¹ s⁻²

• • When travelling from 3 Earth radii to one Earth radii the potential difference is: −4.16 × 10⁷ J kg⁻¹

• The potential difference is defined as:

The work done by moving a positive test charge from one point to another in an electric field

• The units for potential difference are:
  • Joules per Coulomb for electrostatic potential difference
  • Joules per Kilogram for gravitational potential difference
• This can be related to the concept of equipotentials where the movement of a small test mass or positive test charge from one equipotential to another visually represents a potential difference
  • Further, potential difference also occurs across electrical components which is also known as voltage

Uniform Electric Field Strength

• The magnitude of the electric field strength in a uniform field between two charged parallel plates is defined as:

• Where:
• E = electric field strength (V m^-1)
• V = potential difference between the plates (V)
• d = separation between the plates (m)

• Note: the electric field strength is now also defined by the units V m^-1
• The equation shows:
• The greater the voltage between the plates, the stronger the field
• The greater the separation between the plates, the weaker the field

• This equation cannot be used to find the electric field strength around a point charge (since this would be a radial field)
• The direction of the electric field is from the plate connected to the positive terminal of the cell to the plate connected to the negative terminal

The E field strength between two charged parallel plates is the ratio of the potential difference and separation of the plates

• Note: if one of the parallel plates is earthed, it has a voltage of 0 V

Derivation of Electric Field Strength Between Plates

• When two points in an electric field have a different potential, there is a potential difference between them
  • To move a charge across that potential difference, work needs to be done
  • Two parallel plates with a potential difference ΔV across them create a uniform electric field

The work done on the charge depends on the electric force and the distance between the plates

• Potential difference is defined as the energy, W, transferred per unit charge, Q, this can also be written as:

• Therefore, the work done in transferring the charge is equal to:

• Where:
• W = work done (J)
• F = force (N)
• d = distance (m)

• Equate the expressions for work done:

• Since  the electric field strength between the plates can be written as:

• • Potential difference, V = 7.9 kV = 7.9 × 10³ V
  • Distance between plates, d = 3.5 cm = 3.5 × 10⁻² m
  • Charge, Q = 2.6 × 10⁻¹⁵ C

Step 4: Substitute the values into the electric force equation

= 5.869 × 10⁻¹⁰ N

• • The magnitude of the electric force acting on this charged particle is 5.9 × 10⁻¹⁰ N