Capacitors

• Unit of capacitance is the farad (F)
• Capacitance is the charge an object can store per unit potential difference across it

  
• A capacitor is made up of two conducting parallel plates with a dielectric in between them
• When the capacitor is connected to a power source, opposite charges build up on the two parallel plates causing a uniform electric field to form
• To increase external force, increase emf, as this increases field strength which increases force
• A = area

  d = distance between plates

  ε0 = permittivity of free space

  εr = permittivity of dielectric

  A) A

  B) d

  
• permittivity = εr x ε0
• Permittivity is a measure of the ability to store an electric field in the material
• relative permittivity can be calculated by finding the ratio of the permittivity of the dielectric to the permittivity of free space

  A) r

  B) 0

  
• Relative permittivity, or dielectric constant, can be though of as the factor by which the dielectric increases the capacitance
• If you increase area you increase capacitance as electrons can spread out and more can fit on plates
• A larger distance between the plates means the electric field between them is weaker, reducing the ability of the capacitor to store charge
• Dielectrics can made of polar molecules
• Polar molecules have one end that is positive and one that is negative
• When polar molecules are unpolarised they lie in random directions. They align (polarise) when an electric field is applied, negative ends pointed towards the positive plate.

  This polarisation produces an internal electric field (each molecule has its own electric field) which opposes the electric field produced by the charges on the plates, reducing this field.

  TF reducing the potential difference required to charge the capacitor decreases, causing capacitance to increase as C = Q/V
• The electrical energy stored by a capacitor is given by the area under a graph of charge against potential difference

  
• The graph of charge against potential difference forms a straight line through the origin
• Energy stored by a capacitor = 1/2 x charge x potential difference
• Three versions of the equation for energy stored by a capacitor

  A) V

  B) Q

  C) Q

  D) C

  
• Work done is equal to the energy stored by the power supply. W = QV. The first electrons require less work, each needs more until the final electrons work is the same as the power supply (QV)
• If the dielectric is removed:

  • Connected to a power supply.
  • V constant
  • C decreases
  • TF Q decreases
  • Isolated
  • Q constant as electrons can't go anywhere
  • C decreases
  • TF V increases
• If the dielectric is removed (energy):

  • Connected to a power supply.
  • V constant
  • C decreases
  • TF W decreases
  • Isolated
  • Q constant as electrons can't go anywhere
  • C decreases
  • TF W increases -> as work has to be done to take the dielectric out
• In order to charge a capacitor, you have to connect it in a circuit with a power supply and a resistor
• Charging a capacitor:

  A) I

  
• Charging a capacitor:

  • asymptote at V0 = terminal p.d. of the battery

  A) V

  
• Charging a capacitor:

  • max charge = Q0

  A) Q

  
• POTENTIAL DIFFERENCE

  Charging a capacitor:

  • Q=CV, TF pd is proportional to charge
  • As Q builds up so does pd
  • Max. value of pd reached is equal to terminal pd of the battery
• CHARGE

  Charging a capacitor:

  • Charge stored by the capacitor increases with every electron that moves from the positive to the negative plate
  • The charge increases quickly at the beginning because a large current is flowing
  • As the current drops, the rate at which the charge increases also drops
  • A maximum charge is reached
• CURRENT

  Charging a capacitor:

  • Current = flow of electrons
  • Large current initially as very little work required to move electrons onto/off a plate
  • Overtime, more work is needed due to electrostatic repulsion/attraction of similar/opposite charges, so current falls
  • As work required to move electron onto/off a plate approaches eV, the current falls to zero
• Charging a capacitor:

  • Capacitor connected to a power supply and current starts to flow
  • Negative charge builds up on plate connected to negative terminal
  • On the opposite plate, electrons are repelled by the negative charge building up on the initial plate, electrons move to the positive terminal
  • Equal but opposite charge is formed on each plate = potential difference
  • As charge across plates increases, pd increases but electron flow decreases due to the force of electrostatic repulsion increasing
  • TF current decreases and eventually reaches zero
• To discharge a capacitor you must connect it to a closed circuit with just a resisitor
• Discharging a capacitor

  A) I

  B) V]

  C) Q

  
• CURRENT

  Discharging a capacitor:

  • initially a large current as the electrons leave the negative plate
  • As the number of electrons on the plate decreases so does the size of the repulsive electrostatic force
  • This makes the current fall as a slower rate
  • When no more electrons move into the circuit, the current drops to zero
• CHARGE

  Discharging a capacitor:

  • the charge that was stored on the plates now falls with every electron that leaves the negative plate
  • the charge falls quickly initially and then slows, eventually reaching zero when all the charge has left the plates
• POTENTIAL DIFFERENCE

  Discharging a capacitor:

  • As the charge falls to zero so does the potential difference across the capacitor
• Discharging a capacitor:

  • The current flows in the opposite direction, and the current, charge and pd will all fall exponentially, meaning it will take the same amount of time for all the values to halve
• R is the resistance of the charge/discharge circuit

  A) Charging

  B) Discharging

  
• DISCHARGING

  The time constant (RC) is the time it takes for the charge, pd and/or current to fall to 37% of its original value
• CHARGING

  The time constant (RC) is the time it takes for the charge or pd to rise to 63% of its final value, or for the current to fall to 37% of its original value
• A capacitor can be considered fully charged or discharged after 5 time constants.

  • Charging -> charged to over 99% of final value
  • Discharging -> have less than 1% of initial charge remaining