Module 6: Chapter 21 - Capacitance Flashcards
What is the capacitance equation?
C = Q/V
What is a capacitor?
An electrical component that stores charge, consisting of two plates separated by an insulator (a dielectric)
What is the structure of a capacitor?
What is the circuit symbol for a capacitor?
What is capacitance?
The charge stored per unit potential difference difference across a capacitor
What is the unit for capacitance?
Farads (F)
Why is the charge stored in a capacitor directly proportional to the potential difference across it?
Q = VC. The charge is always directly proportional to the potential difference as for a greater build up of positive and negative charge stored on the 2 plates, the greater the p.d across them
Explain how a capacitor charges up
When the capacitor is connected to the cell, electrons flow from the cell for a very short period of time. They cannot travel between the plates because of the insulation. The very brief current means electrons are deposited on one of the plates of the capacitor (on the same side as the negative terminal of the cell) and electrons are removed from the opposite plate (on the same side as the positive terminal of the cell). The plate that gains electrons acquires a negative charge and the plate that loses electrons becomes electron deficient and acquires a net positive charge. The 2 plates have and equal but opposite charge of magnitude Q, and therefore there is a potential difference across the 2 plates. The current falls to 0 when the p.d across the plates is equal to the e.m.f of the cell - the capacitor is fully charged
What is the net charge on a capacitor?
0C, the capacitor is separating a charge of +Q and -Q, therefore there is no overall charge
When is a capacitor fully charged?
When the potential difference across the capacitor is equal to the emf of the cell charging it
What is the equation for the total capacitance of capacitors in parallel?
Cₜ = C₁ + C₂ + C₃…
Derive the equation for the total capacitance of capacitors in parallel
- The pd across each capacitor is the same: Vₜ = V₁ = V₂ = V₃
- Electrical charge is conserved, therefore the total charge stored is equal to the sum of the individual charges stored: Qₜ = Q₁ + Q₂ + Q₃
Qₜ/Vₜ = Q₁/V₁ + Q₂/V₂ + Q₃/V₃
Qₜ/Vₜ = Q₁/Vₜ + Q₂/Vₜ + Q₃/Vₜ
Cₜ = C₁ + C₂ + C₃
What is the equation for the total capacitance of capacitors in series?
1/Cₜ = 1/C₁ + 1/C₂ + 1/C₃…
Derive the equation for the total capacitance of capacitors in series
- The total pd across components in series is equal to the sum of the individual pd’s: Vₜ = V₁ + V₂ + V₃
- The charge stored on each capacitor is the same as the current through each capacitor is the same: Qₜ = Q₁ = Q₂ = Q₃
Vₜ/Qₜ = V₁/Q₁ + V₂/Q₂ + V₃/Q₃
Vₜ/Qₜ = V₁/Qₜ + V₂/Qₜ + V₃/Qₜ
1/Cₜ = 1/C₁ + 1/C₂ + 1/C₃
Why do capacitors in series accumulate the same charge?
The same current flows through both capacitors, meaning the number of electrons deposited on the negative plate of each capacitor is equal
Calculate the value of V on the voltmeter reading
2.6V
What can be said about the charge in this circuit?
The total charge stored (after combining the capacitors into a single capacitor) is equal to the charge stored by the 100μF capacitor which is equal to the charge stored by the parallel capacitors together (each parallel plate stores half of this charge)
Explain what would happen when the switch is closed
When the switch is closed, there is a maximum current in the circuit and the capacitor begins to charge up. the potential difference across the capacitor starts to increase from zero as it gathers charge. According to kirchoff’s second law, the pd Vᵣ across the resistor and the pd Vc across the capacitor must always add up to V₀, therefore Vᵣ decreases as Vc increases. After a long time (depending on the time constant of the circuit), the capacitor will be fully charged with a pd of V₀. At this point the potential difference Vᵣ is 0 and the current within the circuit is 0
What circuit set up can be used to confirm the capacitance rules (combinations in series and parallel)?
Explain why work must be done to charge a capacitor
- As an electron is moving towards the negative plate of a capacitor being charged, it will experience a repulsive electrostatic force from all the electrons that are already on the plate. Therefore external work must be done to push this electron onto the negative plate and charge the capacitor
- Similarly, as an electron is moving away from the positive plate, it will experience an attractive electrostatic force. Therefore, external work must be done to cause the electron to leace the positive plate of the capacitor
What provides the external work to charge a capacitor?
The energy stored in the battery or power supply
How do you determine the energy stored in a capacitor from a pd-charge graph of the capacitor?
The area under the graph
What is the shape of a pd-charge graph of a capacitor?
Linear - potential difference across a capacitor is always directly proportional to the charge stored
What are the 3 equations for the energy stored in a capacitor?
- W = 0.5 QV
- W = 0.5 V²C
- W = 0.5 Q²/C
Derive the equation W = 0.5 QV
Work done = area under the graph
W = 0.5 x ΔV x ΔQ
W = 0.5 QV
Derive the equation W = 0.5 V²C
W = 0.5 VQ
W = 0.5 V(VC)
W = 0.5 V²C
Derive the equation W = 0.5 Q²/C
W = 0.5 QV
W = 0.5 Q(Q/C)
W = 0.5 Q²/C
What is exponential decay?
A constant-ratio process in which a quantity decreases by the same factor in equal time intervals
What can be said about the potential difference across a capacitor discharging through a resistor?
It decreases (decays) exponentially over time
The pd across the battery is V₀, the capacitor is charged until the potential difference across it is V₀. The switch is then opened, explain what happens:
- At time t=0, the pd across the capacitor is V₀ and the charge stored on it is V₀C (as Q = VC). For the resistor, the current through it is V₀/R (as V = IR).
- The capacitor then discharges through the resistor. The charge stored on the capacitor decreases over time and hence the pd across it also decreases. Therefore, the current in the resistor decreases as potential difference across the capacitor decreases. Eventually, the pd, the charge stored, and the current all fall to zero
What is the equation for the potential difference across a capacitor discharching
V₀ = pd at t=0, C = capacitance, R = resistance of circuit being discharged through, t = time
What is the equation for the current through a capacitor discharching
I₀ = current at t=0, C = capacitance, R = resistance of circuit being discharged through, t = time
What is the equation for the charge stored in a capacitor discharching
Q₀ = charge stored at t=0, C = capacitance, R = resistance of circuit being discharged through, t = time
What is similar between p.d, charge, and current when a capacitor is discharging?
They all show exponential decay over time with the general relationship:
Where x = V, I, or Q
Draw the V-t graph for a discharging capacitor:
Draw the I-t graph for a discharging capacitor:
Draw the Q-t graph for a discharging capacitor:
What is the constant-ratio property of exponential decay in respect to a discharging capacitor?
The factor (p.d, charge, or current) decreases by a constant factor in equal time periods, i.e, if x₁, x₂, and x₃ each have a separation of 20 seconds: x₁/x₂ ≈ x₂/x₃
What is constant property of a capacitor discharging?
Capacitance
What is the time constant (τ) of a capacitor-resistance circuit?
The time constant (τ) of a capacitor-resistor circuit is equal to the product of capacitance and resistance (CR)
What does the time constant (τ) represent?
The time constant (τ) acts as a useful measure for how long the exponential decay will take in a particular capacitor-resistor circuit. It is equal to the time taken for the p.d (or current or charge) to decrease to e⁻¹ (about 37%) of its initial value
Prove the time constant (τ) is a measure of time?
The time constant (τ) is equal to CR. By using dimensional analysis you can show that this is equal to time:
FΩ
= C/V x V/A
= As/A
= s
Explain how the equation ΔQ/Δt = -Q/CR can be used to model the decay of charge Q on the capacitor
It can be used to model the decay of charge on the capacitor using a technique called iterative modelling:
1. Start with a known value for the initial charge Q₀ and a known value for the time constant CR
2. Choose a time interval Δt which is very small compared with the time constant
3. Calculate the charge leaving the capacitor, ΔQ, in a time interval Δt using the equation: ΔQ = Δt/CR x Q
4. Calculate the charge Q left on the capacitor at the end of the period Δt by subtracting ΔQ from the previous charge
5. Repeat the whole process for the subsequent multiples of the time interval Δt
Derive the equation “ΔQ/Δt = -Q/CR”
Charge stored by the capacitor: Q = VC
Current in circuit: I = V/R
Since V is constant in the circuit: I = Q/CR
For a capacitor I = -ΔQ/Δt
ΔQ/Δt = -Q/CR
What is the equation used to model the decay of charge on the capacitor?
ΔQ/Δt = -Q/CR
ΔQ is the charge lost in each time interval Δt
What is the equation for the current in the circuit when a capacitor is charging?
I₀ = maximum current when t=0, C = capacitance, R = resistance of circuit, t = time
What is the equation for the potential difference across the resistor when the capacitor is being charged?
V₀ = maximum potential difference when t=0, C = capacitance, R = resistance of circuit, t = time
How does the current in the circuit change as a capacitor is being charged?
It decreases exponentially with respect to time
How does the potential difference across the resistor change as the capacitor is being charged?
It decreases exponentially with respect to time
Draw a graph for the potential difference across the resistor and the potential resistance across the capacitor over time as the capacitor is being charged
What is the equation for the potential difference across the capacitor as it is being charged?
Derive the equation
V₀ = Vᵣ + Vc
Vc = V₀ - Vᵣ
Vc = V₀ - V₀e^(-t/CR)
Vc = V₀(1 - e^(-t/CR) )
What is the equation for the charge stored on the capacitor as it is being charged?
Be careful of whether the capacitor is charging or discharging!!!
What is the difference between the energy released by a capacitor and the energy released by a chemical cell?
- Unlike chemical cells, capacitors cannot store a great deal of energy in such a small volume
- However, capacitors can release the stored energy very quickly and thus have a much higher power output
Why does the current in a circuit exponentially decrease as a capacitor is charging?
As the capacitor is charging, it becomes more and more difficult to remove electrons from the positive plate and more and more difficult for electrons to flow onto the negative plate. Therefore, the rate of flow of charge (current) falls exponentially with time
What is the potential difference across the capacitor when the capacitor is fully charged?
Vc = V₀
A capacitor is charged through a fixed resistance of 100kΩ. If the power supply has an emf of 6.0V and the capacitor has a capacitance of 500μF, what will be the charge stored on the capacitor after 20s?
9.89x10⁻⁴ C
What is the power output of a capacitor?
Power output = Energy stored / time to discharge
What are 4 uses of capacitors?
- Smoothing an output voltage (rectifier circuits)
- Camera flashes
- Emergency lights
- Computer emergency back up supply
How are capacitors used in camera flashes?
Camera flashes have large power outputs (bright flashes) by discharging their capacitors in a very short period of time
How are capacitors used in emergency lights?
Emergency lights would discharge their capacitors over a longer period of time so that back up power could be supplied until normal power has been restored (high time constant)
How are capacitors used in computer emergency back up supplies?
Computers will have an emergency back up supply using capacitors that wil allow them to close down properly in the event that the battery supply terminates
How are capacitors used in rectifier circuits?
Domestic electricity is supplied as an alternating current from the national grid (as seen in the first diagram). A rectifier circuit changes an alternating voltage into a smooth direct voltage. The diode in the circuit allows the current to flow in one direction only. Without the capacitor, the output voltage from the circuit would consist of positive cycles of the ac voltage only - resulting in a massively varying voltage (as seen in the second diagram). With the capacitor, the output voltage is smoothed out and becomes almost a direct voltage of a constant value (as seen in the third diagram). By making the time constant of the circuit very large (larger than the frequency of the ac supply) you reduce the ripple in the output voltage
How can you reduce the ripple in the output voltage of a rectifier circuit?
Ensure that the time constant is much larger than the frequency of the AC supply
How can you determine the time constant, CR, of a circuit using a voltmeter and a stopwatch?
- Discharging Method: Close the switch and ensure the capacitor is fully charged. Connect the voltmeter across the capacitor and record the initial value. As you open the switch start the stopwatch. Stop the stopwatch once the reading on the voltmeter has reached 37% of it initial reading (37% of V₀)
- Charging Method: Open the switch and ensure the capacitor is fully discharged. Connect the voltmeter across the resistor and record the initial value. As you close the switch start the stopwatch. Stop the stopwatch once the reading on the voltmeter has reached 37% of it initial reading (37% of V₀)
An uncharged capacitor of capacitance C is charged through a resistor of resistance R using a battery of emf V₀. Derive an equation for the time t taken for the pd across the capacitor and the resistor to be the same
A 500μF capacitor is discharged through a 200kΩ resistor. Calculate the time taken for the current in the resistor to decrease to 25% of its initial value
q
140s
What must you be very careful about in capacitance calculations?
Whether the capacitor is charging or discharging
