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

Step 2: Write down the equation for potential differenceExam Tip

• 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:Worked Example

• • 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 2: Write down the equation for the electric field strength between the parallel platesStep 5: State the final answer

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