Capacitors

• Electric field lines between plates are straight and evenly spaced -> uniform electric fields.

  
  

  Electric field has a strength (symbol E, measured in Vm$^-$$^1$) that depends on the voltage across the plates (V) and the distance between the plates (d). They’re connected by this equation: E = V/d

  
• Capacitance story part 1

  The capacitor is a break in the circuit, current can flow onto metal plates but no electrons. When battery is turned on current flows and electrons gather on the right plate. Charge of -Q builds on right plate. Charge +Q builds on the left plate. As charge flows onto the plates a uniform electric field grows between them. So a voltage grows between plates too. When this voltage across capacitor is exactly equal to source voltage, current dies -> charging of capacitor stops when Vc = V.
• Capacitance story part 2


  Capacitance is the amount of charge that can be stored on the capacitor divided by voltage across the plates: C = Q/V
• Capacitance story part 3

  A battery charges the capacitor by pushing electrons onto the plates. This stops when the voltage across the plates is equal to the source voltage. The capacitance is how much charge (on the positive plate) there is divided by the source voltage when charging is done

  capacitance has symbol C for coulomb measured in Farads F. (usually micro farads or nano farads)
• Maximise capacitance part 1

  Increasing area of plates would increase the capacitance because more charge can flow onto the plates for the same voltage.

  
  

  Equation: C = (ε0 εr A)/d

  
• Maximise capacitance part 2

  Decreasing plate separation (d) also increases capacitance.

  Equation: C = Q/V (capacitance is how much charge can be stored for a given voltage)

  
  

  V = Ed -> a decrease in d leads to a decrease in

  V. They’re proportional.

  
  

  Decrease in V leads to an increase in C. Inversely proportional. So to increase capacitance (C) decrease distance (d)
• Maximise capacitance part 3

  Capacitance increases with dielectric in between plates -> causes Efield between plates to pull -ve sides of dielectric molecules to +ve plate and +ve sides of molecule to -ve plate.

  
  

  When positive end of dielectric molecules touch negative electrons in negative plate some cancelling happens, reducing amount of effective charge on negative plate, same amount of cancelling happens at positive plate.

  
  

  Charge on two plates drop, more charge can flow on plates without changing source voltage. More charge for same voltage gives better capacitance, since C = Q/V.

  
• Summary

  There are 3 ways to increase the capacitance of a parallel plate capacitor:

  • Increase the plate area (A).
  • Decrease the plate separation (d).
  • Place a dielectric material (εr) between the plates
• Energy

  When charged up, capacitors store energy. Depends on how much charge is on the plates(Q) and the voltage keeping that charge there(V). This energy is electric potential energy. If the battery is turned off, the energy will flow off the capacitor plates and become kinetic energy. 

  
  

  Formula: E = 1/2 Q x v

  Formula for charge Q = C x V
• Combining capacitors

  Capacitances in series: 1/CT = 1/C1 + 1/C2 + 1/C3 ...

  Capacitances in parallel: CT = C1 + C2 + C3 ...

  
• RC circut

  
• Charging capacitors part 1

  Capacitor starts with no charge. When switch

  closes, there is zero charge on the capacitor and so zero voltage across it (because the voltage across the capacitor is VC = Q/V). VC = 0. There is, a voltage due to the battery VS.

  
  

  RC circuit: Vs - VR = Vc = 0 (so Vs = VR + Vc).

  But, at the start Vc = 0 so VR = Vs

  
  

  Eventually Vc = Vs. Resistor voltage -> VR = Vs - Vc but Vc = Vs so VR = 0

  So resistor voltage starts at source voltage and drops to 0. Capacitor voltage starts at 0 and rises to source voltage.
• Charging capacitors part 2

  When charging a capacitor, Vc goes from 0(when switch closes) to Vs(fully charged) exponentially.

  VR goes from Vs down to 0 exponentially.

  
• Charging capacitors part 3


  Current during charging process:

  Initial voltage in the whole circuit is the supply voltage Vs -> Ohm's law -> I(i) = Vs/R

  So current starts out as source voltage divided by resistance. When capacitor is fully charged current in circuit stops. So I(f) = 0. Curve for current is exponential.

  
• Discharging capacitors


  Start with fully charged capacitor and no battery. Initially capacitors voltage is Vc = Q/C. And finally capacitor voltage is 0.

  So Vc curve drops for discharging capacitor. Kirchhoff's law: Vc - VR = 0. So, Vc = VR

  VR and Vc curve are the same.

  I = V/R so I curve drops down with same shape, starting at value V/R. V is initial value for voltage curves Q/C

  
• Time constant


  To know when capacitor is fully charged or discharged -> never fully reach zero or maximum, so approximate using time constant (τ).

  Amount of time in s, it takes for the curves to change (rise or drop) by 63%.

  Rule: After (about) five time constants have passed charging or discharging is complete. Time constant of RC circuit: τ = RC