Electric Fields

Concept of Electric Fields

• Electric fields can be described as a field of force; it can move charged particles by exerting a force on them
• A positive charge moves in the direction of the electric field: they gain E_k and lose E_p
• A negative charge moves in the opposite direction of the electric field: they lose E_k and gain E_p
• The electric field of a charge is the space around the charge in which an electric force due to that charge is experienced
• The direction of the field lines show the direction of the field — always from the positive charge to the negative
• A higher density of the lines shows a stronger region of field

Diagrammatic Representation

• Parallel Plates
  
• Points
  

Electric Field Strength

• It is the force per unit positive charge acting at a point; it is a vector
• Its units include:
  • NC^-1
    • E = ^F/_q
    • E is the electric field strength
    • F is the force
    • q is the charge
  • Vm^-1
    • E = ^V/_d
    • V is potential difference
    • d is the distance between plates
• The higher the voltage, the stronger the electric field
• The greater the distance between the plates, the weaker the electric field

Coulomb’s Law

• Any two points charges exert an electrical force on each other that is proportional to the product of the charges and inversely proportional to the square of separation
• F ∝ ^Qq/_r²
• F = ^Qq/_4πε0r²

Electric Field of a Point Charge

• Electric field strength is force per unit positive charge
• Dividing force by charge, q: E = ^Q/_4πε0r²

Electric Potential

• Electric potential at a point is the work done in bringing a unit positive charge form infinity to that point
• W = VQ and W = Fd
• V = ^Fd/_Q
• V = ^Q/_4πε0r
• The potential difference between two points A and B from an isolated charge Q is defined as the work done in taking a unit positive charge from B to A

• V_AB = ^Q/_4πε0(¹/_b – ¹/_a)
• V_AB is equal to the gain in electrical potential energy if Q is positive, and loss if Q is negative
• In general
  • If a +ve is moved in the direction of the electric field, its electric potential energy will decrease
  • If a -ve charge is moved in the direction of the electric field, its electric potential energy will increase
  • If a charge is accelerated in the field, its electric potential energy will be converted to kinetic
    ∴ Vq = ½mv²
    

Examples

1. The maximum field strength at the surface of a sphere before electrical breakdown (sparking) occurs is 2.0 × 10⁶ Vm^-1. The sphere has a radius r of 0.35 m.
   Calculate the maximum values of
   1. the charge that can be stored on the sphere
      • Maximum field strength is given, therefore the fields strength formula should be used:
        • E = ^Q/_4πε0r²
      • Substitute the given information:
        • 2 × 10⁶ = ¹/_4πε0 × ^Q/_0.35²
        • ∴ Q = 2.7 × 10^-5 C
   2. the potential at the surface of the sphere
      • using the potential equation:
        • V = ^Q/_4πε0r
      • substitute the given information:
        • V = ¹/_4πε0 × ^{2.6 × 10-5}/_0.35
        • ∴ V = 7.0 × 10⁵ V

Potential Due to a Conducting Sphere

• A charge +Q on an isolated conducting sphere is uniformly distributed over its surface
  
• The charge remains on the surface and at all points inside the sphere, the field strength is 0
• As there is no field inside the sphere, the potential difference from any point inside the sphere to the surface is zero.
• Therefore, the potential at any point inside a charged hollow sphere is the same as its surface
  

Equipotential

• An equipotential surface is a surface where the electric potential is constant
• Equipotential lines are drawn such that the potential is constant between intervals
• As the potential is constant, the potential gradient = 0, hence E along the surface = 0
• Hence no work is done when a charge is moved along this surface
  
• Electric field lines must meet equipotential surfaces at right angles
• Spacing will be closer when the field is stronger

Similarity and Difference between Electric and Gravitational Potential

• Similarities:
  • Ratio of work done to mass/charge
  • Work done moving a unit mass/charge from infinity
  • Both have zero potential at infinity
• Differences:
  • Gravitational forces are always attractive
  • Electric forces can be attractive or repulsive
  • For gravitational, work gets out as masses come together
  • For electric, work done is on charges if they have the same sign, and work done gets out if opposite charges come together

Capacitance

Capacitors

• Capacitors are used to store energy
• A dielectric is an electrical insulator
• How Capacitors Store Energy:
  • On a capacitor, there is a separation of charge with +ve on one plate and -ve on the other
  • To separate the charges, work must be done, hence energy is released when charges come together

Capacitance and Farad

• Capacitance is the ratio of the charge stored by a capacitor to the potential difference across it
• Farad is the unit of capacitance. It represents 1 coulomb per volt
  • C = ^Q/_V
• The capacitance of a capacitor is directly proportional to the area of the plates and inversely proportional to the distance between the plates

Dielectric Breakdown

• An electric field can cause air to become conducting by:
  • The electric field causes forces in opposite directions on the electrons and the nucleus of atoms in air
  • This results in the field causing electrons to be stripped off the atom
  • Results in a spark — air now contains oppositely charged particles which can carry charge

Capacitors in Parallel

• By conservation of energy and hence charge (W = QV), the total charge in a circuit is the sum of individual charges
  • Q_T = Q₁ + Q₂ + Q₃
• Apply Q = CV and V is constant in parallel:
  • Q_T = V(C₁ + C₂ + C₃)
  • ^Q_T/_V = C₁ + C₂ + C₃
• Hence
  • C_T = C₁ + C₂ + C₃

Capacitors in Series

• The total p.d. in a circuit is the sum of the individual p.d.
  • VT = V₁ + V₂ + V₃
• Apply Q = CV and Q is constant in series:
  • V_T = Q(¹/_C1 + ¹/_C2 + ¹/_C3)
  • ^V_T/_Q = ¹/_C1 + ¹/_C2 + ¹/_C3
• Hence
  • ¹/_CT = ¹/_C1 + ¹/_C2 + ¹/_C3

Capacitance of a Body

• Any isolated body can have a capacitance
• Considering a sphere of radius r, carrying a charge Q, the potential at the surface is
  • V = ^Q/_4πε0r
  • C = ^Q/_V = Q ÷ ^Q/_4πε0r
  • C = 4πε₀r

Examples

1. An isolated metal sphere of radius 63 cm is charged to a potential of 1.2 × 10⁶ V. At this potential, there is an electrical discharge in which it loses 75% of its energy
   1. Calculate the capacitance of the sphere
      • Using the equation derived above:
        • C = 4π × 8.85 × 10^-12 × 63 × 10^-2
        • ∴ C = 7.0 × 10^-11 Farad
   2. Calculate the potential of the sphere after the discharge has taken place
      • Using the equation for energy:
        • W = CV²
      • After the discharge, the sphere contains 25% of the energy before, so equating the energy before and after:
        • 25% × C × (1.2 × 10⁶)² = CV²
      • Cancel C and calculate V:
        • V = 6.0 × 10⁵ V

Energy Stored in a Capacitor

• Area under a potential-charge graph is equal to the work done
  • W = ½QV = ½CV²
• The half comes in because:
  • When the first charge flows onto the capacitor plates, there is no potential difference opposing the flow
  • As more charges flow, the potential difference increases, so more work is done
  • The average potential difference is equal to half the maximum potential difference