7.4 capacitance

• Capacitor 

  • An electrical component made up of two conducting plates separated by a gap or a dielectric
  • When a capacitor is connected to a power source, positive and negative charge build up on opposite plates, creating a uniform electric field between them
  •  Capacitance = charge per unit potential difference stored by a capacitor
  • $C=$$\frac{Q}{V}$
  • Measured in farads
• Dielectrics

  • Insulating material used to reduce electric field
  • Permittivity - the measure of how difficult it is to generate an electric field in a certain material
  • Relative permittivity - ratio of the permittivity of a material to the permittivity of free space
  • Dielectric constant
  • $\epsilon_r=$$\frac{\epsilon_1}{\epsilon_0}$
• ... dialectrics

  • No charge applied
  • Made up of polar molecules - positive and negative ends
  • When no charge is stored, no electric field - molecules point in random directions
• ...dialectrics

  • Charge applied
  • When a charge applies, an electric field is generated
  • Negative ends attracted to the positively charged plate and vice versa
  • All molecules rotate and align themselves with the electric field
  • Molecules have their electric field opposing the applied field - the larger the permittivity, the larger the opposing field
  • Reduces overall electric field, reducing the potential difference needed to charge the capacitor - capacitance increases
  • $C=$$\frac{A\epsilon_0\epsilon_r}{d}$

  
• Storing energy

  • Electric energy is stored when charge builds up on the plates of the capacitor
  • Area under charge-potential difference graph 
  • Potential difference across the capacitor is directly proportional to the charge stored on it 
  • straight line through the origin
  • The greater the capacitance, the more energy stored
  • $E=$$\frac{1}{2}QV\ \ \ \ \ E=$$\frac{1}{2}CV^2\ \ \ \ \ E=$$\frac{1}{2}\times\frac{Q^2}{C}$  

  
• Time constant, RC 

  • Time taken to charge or discharge a capacitor depends on capacitance and resistance
  • Capacitance affects the amount of charge transferred at a given potential difference
  • Resistance affects the current
• ... time constant

  • When $t=$$\frac{1}{2.718}\approx0.37$
  • Time taken for charge, potential difference or current of discharging the capacitor to fall to 37% of its value when fully charged
  • Taken for charge or potential difference of charging the capacitor to rise to 63% of its value when fully charged
  • The larger the resistance in series with the capacitor, the longer it takes to charge or discharge
  • Time taken for a capacitor to charge/discharge fully is approximately 5RC
• Calculation of time constant

  • $Q=$$Q_0e^{-\frac{t}{RC}}$
  • Works for current and potential difference
  • Take natural logs of both sides
  • $\ln\left(Q\right)=$$\left(-\frac{1}{RC}\right)t+$$\ln\left(Q_0\right)\$
  • Now in the form of a straight line
  • Gradient of this line is $-\frac{1}{RC}$
  • Y-intercept is $\ln\left(Q_0\right)$
  • Find time constant -1 $\div$ gradient
• Time to halve

  • Time taken for the charge, current, or potential difference of a discharging capacitor to reach half of the value it was when fully charged

  $T_{\frac{1}{2}}=$$0.69RC$
• Discharging capacitors

  • Charge left on the plates falls exponentially with time 
  • As the capacitor discharges, it always takes the same length of time for the charge to halve regardless of the initial value
  • Charge left on the plate after the given time
  • $Q=$$Q_0e^{-\frac{t}{RC}}$
  • As the potential difference and the current also decrease exponentially as the capacitor discharges, the equations are similar
  • $I=$$I_0e^{-\frac{t}{RC}}$
  • $V=$$V_0e^{-\frac{t}{RC}}$

  
• Charging capacitors

  • When a capacitor is charging, the growth rate of charge and the potential difference across the plate shows exponential decay
  • Charge on the plate at the given time after the beginning of charging
  • $Q=$$Q_0\left(1-e^{-\frac{t}{RC}}\right)$
  • Potential difference between plates at given time
  • $V=$$V_0\left(1-e^{-\frac{t}{RC}}\right)$
  • Charging current decreases exponentially
  • $I=$$I_0e^{-\frac{t}{RC}}$

  
• Core practical 9: Charging and discharging capacitors 

  • Set up circuit
  • Close switch to connect uncharged capacitor to dc power supply
  • Let the capacitor charge and record both the potential difference and the current over time
  • When the current is zero, the capacitor is fully charged
• ... practical: charging capacitors

  • When circuit is closed, current flows
  • electrons flow onto plate connected to negative terminal of dc power supply, so negative charge builds up
  • Negative charge repels electrons off the plate connected to the positive terminal of the power supply, making that plate positive, and these electrons are attracted ot the positive terminal
• ...practical: charging capacitors

  • Equal and opposite charge builds up on each plate, causing a potential difference between the plates
  • No charge can flow between the plates as they're separated by an insulator
  • Charge builds up on the plates, increasing electrostatic repulsion, making it harder for more electrons to be deposited
  • When the pd across the capacitor is equal to the pd across the power supply, the current falls to zero
• ...practical: discharging capacitors

  • Remove power supply and close switch to complete circuit
  • Let capacitor discharge and record potential difference / current over time
  • When current and potential difference are zero, capacitor is fully discharged 
  • Current flows in opposite direction from charging current
  • Potential difference increases as current decreases
  • When capacitor is discharging, charge and potential difference decrease exponentially with time
  • Always takes the same length of time to halve regardless of starting value
  • Same for the amount of current flowing around the circuit