The Discharge Equation

• When a capacitor discharges through a resistor, the charge stored on it decreases?exponentially
• The amount of charge remaining on the capacitor?Q?after some elapsed time?t?is governed by the?exponential decay equation:

• 
• ?Where:
  • Q?= charge remaining (C)
  • Q0?= initial charge stored (C)
  • e?= exponential function
  • t?= elapsed time (s)
  • R =?circuit?resistance (Ω)
  • C?= capacitance (F)

Discharge Equation for Potential Difference

• The exponential decay equation for?charge?can be used to derive a decay equation for potential difference
• Recall the equation for charge?Q?=?CV
  • It also follows that the initial charge?Q0?=?CV0?(where?V0?is the initial potential difference)

• Therefore, substituting?CV?for?Q?into the original exponential decay equation gives:

• 
• Cancelling?C?from both sides gives the exponential decay equation for?potential difference?V:

• 
• ?Where:
  • V?= potential difference after some time?t?(V)
  • V0?= initial potential difference (V)
  • t?= elapsed time (s)
  • R?= resistance (Ω)
  • C?= capacitance (F)
• This equation shows that the potential difference also decreases?exponentially,?from some initial value?V0

• ---

Discharge Equation for Current

• The exponential decay equation for?potential difference?can be used to derive a decay equation for current
• Recall Ohm's law?V?=?IR
  • It follows that the initial potential difference?V0?=?I0R?(where?I0?is the initial current)

• Therefore, substituting?IR?for?V?into the decay equation for potential difference gives:

• 
• Cancelling?R?from both sides gives the exponential decay equation for?current?I:

• 
• ?Where:
  • I?= current after some time?t?(A)
  • I0?= initial current (A)
  • t?= elapsed time (s)
  • R?= resistance (Ω)
  • C?= capacitance (F)

• This equation shows that the current also decreases?exponentially,?from some initial value?I0

Step 1: Write the known quantities

• • Initial potential difference?V0?= 12 V
  • Resistance?R?= 1 kΩ = 1000 Ω
  • Capacitance?C?= 10 mF = 0.01?F
  • Time elapsed = 15 s

Step 2: Determine the initial current?I0

• • Since the initial potential difference is 12 V and the resistance is 1000 Ω, then:
• 

Step 3: Write the decay equation for current

• • The decay equation for current is:

Step 4: Substitute quantities and calculate the current after 15 s

• • Substituting quantities gives the following:

I?= (0.012) × (e–(15/(1000 × 0.01))

I?= (0.012) × (e–1.5)

I?= (0.012) × (0.223...)

I?= 2.7 × 10–3?A = 2.7 mA

Natural Logarithms & Discharge Equations

• The exponential decay equations are not?linear
• They can be turned into linear equations by using the?natural logarithm?function
• Recall the exponential decay equation for charge:

• Dividing both sides by?Q0?gives:

• Taking the?natural?logarithm?of both sides 'cancels' the exponential function?e, giving:

• This simplifies to:

• Leaving an equation for the natural logarithm of charge?Q?as:

• This is the equation of a?straight?line graph, where:
  • ln?Q?is plotted on the?y-axis
  • t?is plotted on the?x-axis
  • The gradient of the line is therefore equal to –1/RC

The natural logarithm of the exponential decay curve line arises it to a straight-line graph with a gradient equal to –1/RC

• Following similar steps, the linearised versions of the decay equations for?potential?difference?V?is:

• And for current?I?is:

Step 1: Write the equation for the linearised decay equation for potential difference

• • The linearised decay equation for potential difference is given by:

Step 2: Interpret the graph given using the linearised equation

• • The equation says the y-intercept of the straight line is represented by ln?V0

Step 3: Use the y-intercept to determine the initial voltage

• • The y-intercept is equal to 2.1
  • Therefore:

ln?V0?= 2.1

Step 4: Cancel the natural logarithm using the exponential function:

• • Raising both sides using the exponential function?e?'cancels' the natural logarithm
  • This gives:

V0?=?e(2.1)?= 8.2 V