Charging Curves

• Capacitors are charged by a?power supply?(e.g. a battery)
• When charging, electrons are 'pulled' from the plate connected to the positive terminal of the power supply
  • Hence the plate nearest the positive terminal is?positively charged

   

• Oppositely, electrons are 'pushed' onto the plate connected to the negative terminal
  • Hence the plate nearest the negative terminal is?negatively charged

   

• As the negative charge builds up, fewer electrons are pushed onto the plate due to electrostatic repulsion from the electrons already on the plate
• When no more electrons can be pushed onto the negative plate, the charging stops

A parallel plate capacitor is made up of two conductive plates with opposite charges building up on each plate

• At the start of charging, the current is large and gradually falls to zero as the electrons stop flowing through the circuit
  • The current decreases?exponentially
  • This means the rate at which the charge decreases is proportional to the amount of charge it has left

   

• Since an equal but opposite charge builds up on each plate, the potential difference between the plates slowly increases until it is the same as that of the power supply
  • Therefore, the charge stored on the capacitor plates increases until the potential difference across the plates matches that of the power supply

Graphs of variation of current, p.d and charge with time for a capacitor charging through a battery

• The key features of the charging graphs are:
  • The shapes of the p.d. and charge against time graphs are identical
  • The current against time graph is an?exponential decay?curve
  • The initial value of the current starts on the y axis and decreases exponentially
  • The initial value of the p.d and charge starts at 0 up to a maximum value

   

Discharging Curves

• Capacitors are?discharged?through a resistor with?no?power supply present
• The electrons now flow back from the negative plate to the positive terminal of the power supply until there is potential difference across the capacitor plates
• Charging and discharging is commonly achieved by moving a switch that connects the capacitor between a power supply and a resistor

The capacitor charges when connected to terminal P and discharges when connected to terminal Q

• At the start of discharge, the current is?large?(but in the opposite direction to when it was charging) and gradually falls to zero
• As a capacitor discharges, the current, p.d and charge all decrease?exponentially
  • This means the rate at which the current, p.d or charge decreases is proportional to the amount of current, p.d or charge it has left

   

• The graphs of the variation with time of current, p.d and charge are all identical and follow a pattern of?exponential decay

Graphs of variation of current, p.d and charge with time for a capacitor discharging through a resistor

• The key features of the discharge graphs are:
  • The shape of the current, p.d. and charge against time graphs are identical
  • Each graph shows exponential decay curves with decreasing gradient
  • The initial values (typically called?I0,?V0?and?Q0?respectively) start on the y axis and decrease exponentially

   

• The rate at which a capacitor discharges depends on the?resistance?of the circuit
  • If the resistance is?high, the current will decrease?more slowly?and charge will flow from the capacitor plates more slowly, meaning the capacitor will take?longer?to discharge
  • If the resistance is?low, the current will decrease?quickly?and charge will flow from the capacitor plates quickly, meaning the capacitor will discharge?faster

   

• The time constant of a capacitor discharging through a?resistor?is a measure of how long it takes for the capacitor to discharge
• The definition of the time constant for a?discharging?capacitor?is:

The time taken for the charge, current or potential difference of a discharging capacitor to decrease to 37% of its original value

• Alternatively, for a?charging?capacitor:

The time taken for the charge or potential difference of a charging capacitor to rise to 63% of its maximum value

• 37% is 0.37 or 1 / e (where e is the exponential function) multiplied by the original value (I0,?Q0?or?V0)
• This is represented by the Greek letter tau, , and measured in units of?seconds?(s)
• The time constant provides an easy way to compare the rate of change of similar quantities eg. charge, current and p.d.
• It is defined by the equation:

• Where:
• • = time constant (s)
  • R?= resistance of the resistor (?)
  • C?=?capacitance?of the capacitor (F)

• For example, to find the time constant for a?discharging?capacitor:
• • Calculate 0.37V0,?where?V0?is the?initial?potential difference across it
  • Determine the corresponding time taken for the potential difference to decrease to that value

• To find the time constant for a?charging?capacitor:
  • Calculate 0.63V0,?where?V0?is the?maximum?potential difference across it
  • Determine the corresponding time taken for the potential difference to rise to that value

The time constant shown on a charging and discharging capacitor