# Modelling Capacitor Discharge(OCR A Level Physics)

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Katie M

Expertise

Physics

## Exponential Decay Graph for Capacitors

• To verify if potential difference, V, or charge, Q, on a capacitor decreases exponentially:
• Constant ratio method: Plot a V-t graph and check the time constant is constant, or check if the time to halve from its initial value is constant
• Logarithmic graph method: Plot a graph of ln V against t and check if a straight line graph is obtained

#### Constant Ratio Method

• A general form of the exponential decay question is given by

• Where A is a constant
• This equation shows that when t = A−1 the value of x will have decreased to approximately 37% of its original value, x0:

• Comparing this to the discharge equation for a capacitor:

• Therefore, for a discharging capacitor, when t = τ the potential difference on the capacitor will have decreased to approximately 37% of its original value
• This means that equal intervals of time give equal fractional changes of  in potential difference

The graph of voltage-time for a discharging capacitor showing the positions of the first three time constants

• Hence, to validate if potential difference across a capacitor decreases exponentially:

The time constant, or the time taken for the potential difference to decrease to 37% of its original value, will be constant

• To find the time constant from a voltage-time graph, calculate 0.37V0 and determine the corresponding time for that value

The time constant shown on a charging and discharging capacitor

#### Logarithmic Graph Method

• The potential difference (p.d) across the capacitance is defined by the equation:

• Where:
• V = p.d. across the capacitor (V)
• V0 = initial p.d. across the capacitor (V)
• t = time (s)
• e = exponential function
• R = resistance of the resistor (Ω)
• C = capacitance of the capacitor (F)
• Rearranging this equation for ln(V) by taking the natural log (ln) of both sides:

• Comparing this to the equation of a straight line: y = mx + c
• y = ln (V)
• x = t
• c = ln (V0)

A straight-line logarithmic graph of ln V against t can be used to verify an exponential relationship

#### Worked example

A student investigates the relationship between the potential difference and the time it takes to discharge a capacitor. They obtain the following results:

The capacitor is labelled with a capacitance of 4200 µF.

Calculate the value of the capacitance of the capacitor discharged.

Step 1: Complete the table

• Add an extra column ln(V) and calculate this for each p.d.

Step 2: Plot the graph of ln(V) against average time t

• Make sure the axes are properly labelled and the line of best fit is drawn with a ruler

Step 3: Calculate the gradient of the graph

• The gradient is calculated by:

Step 4: Calculate the capacitance, C

## Modelling the Discharge of a Capacitor

• From electricity, the charge is defined as:

ΔQ = IΔt

• Where:
• I = current (A)
• ΔQ = change in charge (C)
• Δt = change in time (s)

• This means that the area under a current-time graph for a charging (or discharging) capacitor is the charge stored for a certain time interval

The area under the I-t graph is the total charge stored in the capacitor in the time interval Δt

• Rearranging for the current:

• This means that the gradient of the charge-time graph is the current at that time

The gradient of a discharging and charging Q-t graph is the current

• In the discharging graph, this is the discharging current at that time
• In the charging graph, this is the charging current at that time
• To calculate the gradient of a curve, draw a tangent at that point and calculate the gradient of that tangent

• As a capacitor charges or discharges, the current at any time can be found from Ohm's law:

• From the definition of capacitance, the value of potential difference at any time is given by:

• Combining these equations gives:

• For a discharging capacitor, the current decreases with time, hence:

• This leads to an expression which can be used to solve for the time constant of a discharging capacitor:

• This equation is useful for modelling using spreadsheets

#### Worked example

The graph below shows how the charge stored on a capacitor with capacitance C varies with time as it discharges through a resistor.

Calculate the current through the circuit after 4 s.

Step 1: Draw a tangent at t = 4

Step 2: Calculate the gradient of the tangent to find the current I

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