Chapter 8 Capacitance Flashcards
Defining Capacitance
- Capacitors are electrical devices used to store energy in electronic circuits, commonly for a backup release of energy if the power fails
- They can be in the form of:
- An isolated spherical conductor
- Parallel plates
- Capacitors are marked with a value of their capacitance. This is defined as:
The charge stored per unit potential difference
- The greater the capacitance, the greater the energy stored in the capacitor
- A parallel plate capacitor is made up of two conductive metal plates connected to a voltage supply
- The negative terminal of the voltage supply pushes electrons onto one plate, making it negatively charged
- The electrons are repelled from the opposite plate, making it positively charged
- There is commonly a dielectric in between the plates, this is to ensure charge does not freely flow between the plates
A parallel plate capacitor is made up of two conductive plates with opposite charges building up on each plate
Calculating Capacitance
- The capacitance of a capacitor is defined by the equation:
C=Q/V
- Where:
- C = capacitance (F)
- Q = charge (C)
- V = potential difference (V)
- It is measured in the unit Farad (F)
- In practice, 1 F is a very large unit
- Capacitance will often be quoted in the order of micro Farads (μF), nanofarads (nF) or picofarads (pF)
If the capacitor is made of parallel plates
Q is the charge on the plates and V is the potential difference across the capacitor
- The charge Q is not the charge of the capacitor itself, it is the charge stored on the plates or spherical conductor
- This capacitance equation shows that an object’s capacitance is the ratio of the charge on an object to its potential
Capacitance of a Spherical Conductor
- The capacitance of a charged sphere is defined by the charge per unit potential at the surface of the sphere
- The potential V is defined by the potential of an isolated point charge (since the charge on the surface of a spherical conductor can be considered as a point charge at its centre):
V = Q/4πε0r
- Substituting this into the capacitance equation means the capacitance C of a sphere is given by the expression:
C = 4πε0r
Capacitors in Series
- Consider two parallel plate capacitors C1 and C2 connected in series, with a potential difference (p.d) V across them
- In a series circuit, p.d is shared between all the components in the circuit
- Therefore, if the capacitors store the same charge on their plates but have different p.ds, the p.d across C1 is V1 and across C2 is V2
- The total potential difference V is the sum of V1 and V2
V = V1 + V2
- Rearranging the capacitance equation for the p.d V means V1 and V2 can be written as:
V1=Q/C1 and V2=Q/C2
- Where the total p.d V is defined by the total capacitance
V=Q/Ctotal
- Substituting these into the equation V = V1 + V2 equals:
Q/Ctotal=Q/C1 + Q/C2
- Since the current is the same through all components in a series circuit, the charge Q is the same through each capacitor and cancels out
- Therefore, the equation for combined capacitance of capacitors in series is:
1/Ctotal= 1/C1 + 1C2 …
Capacitors in Parallel
- Since the current is split across each junction in a parallel circuit, the charge stored on each capacitor is different
- Therefore, the charge on capacitor C1 is Q1 and on C2 is Q2
- The total charge Q is the sum of Q1 and Q2
Q = Q1 + Q2
- Rearranging the capacitance equation for the charge Q means Q1 and Q2 can be written as:
Q1 = C1V and Q2 = C2V
- Where the total charge Q is defined by the total capacitance:
Q = CtotalV
- Substituting these into the Q = Q1 + Q2 equals:
CtotalV = C1V + C2V = (C1 + C2) V
- Since the p.d is the same through all components in each branch of a parallel circuit, the p.d V cancels out
- Therefore, the equation for combined capacitance of capacitors in parallel is:
Ctotal = C1 + C2 + C3 …
Capacitors connected in parallel have the same p.d across them, but different charge
Capacitors in Series & Parallel
- Recall the formula for the combined capacitance of capacitors In parallel:
Ctotal = C1 + C2 + C3 …
- in series:
Area Under a Potential–Charge Graph
- When charging a capacitor, the power supply pushes electrons from the positive to the negative plate
- It therefore does work on the electrons, which increase their electric potential energy
- At first, a small amount of charge is pushed from the positive to the negative plate, then gradually, this builds up
- Adding more electrons to the negative plate at first is relatively easy since there is little repulsion
- As the charge of the negative plate increases ie. becomes more negatively charged, the force of repulsion between the electrons on the plate and the new electrons being pushed onto it increases
greater amount of work must be done to increase the charge on the negative plate or in other words:
The potential difference V across the capacitor increases as the amount of charge Q increases
As the charge on the negative plate builds up, more work needs to be done to add more charge
The electric potential energy stored in the capacitor is the area under the potential-charge graph
- The charge Q on the capacitor is directly proportional to its potential difference V
- The graph of charge against potential difference is therefore a straight line graph through the origin
- The electric potential energy stored in the capacitor can be determined from the area under the potential-charge graph which is equal to the area of a right-angled triangle:
- area = ½ x base x height
Calculating Energy Stored in a Capacitor
- Recall the electric potential energy is the area under a potential-charge graph
- This is equal to the work done in charging the capacitor to a particular potential difference
- The shape of this area is a right angled triangle
- Therefore the work done, or energy stored in a capacitor is defined by the equation:
W = ½ QV
- Substituting the charge with the capacitance equation Q = CV, the work done can also be defined as:
W = ½ CV2
- Where:
- W = work done/energy stored (J)
- Q = charge on the capacitor (C)
- V = potential difference (V)
- C = capacitance (F)
- By substituting the potential V, the work done can also be defined in terms of just the charge and the capacitance:
- W = Q2/2C
The capacitor charges when connected to terminal P and discharges when connected to terminal Q
Graphs of variation of current, p.d and charge with time for a capacitor discharging through a resistor
The graph of voltage-time for a discharging capacitor showing the positions of the first three time constants
Using the Capacitor Discharge Equation
- The time constant is used in the exponential decay equations for the current, charge or potential difference (p.d) for a capacitor discharging through a resistor
- These can be used to determine the amount of current, charge or p.d left after a certain amount of time when a capacitor is discharging
- The exponential decay of current on a discharging capacitor is defined by the equation:
- Where:
- I = current (A)
- I0 = initial current before discharge (A)
- e = the exponential function
- t = time (s)
- RC = resistance (Ω) × capacitance (F) = the time constant τ (s)
This equation shows what?
- that the faster the time constant τ, the quicker the exponential decay of the current when discharging
- Also, how big the initial current is affects the rate of discharge
- If I0 is large, the capacitor will take longer to discharge
- Note: during capacitor discharge, I0 is always larger than I, this is because the current I will always be decreasing
- The current at any time is directly proportional to the p.d across the capacitor and the charge across the parallel plates
- Therefore, this equation also describes the change in p.d and charge on the capacitor:
equation will sub in V in place of Q for pd
- Where:
- V = p.d across the capacitor (C)
- V0 = initial p.d across the capacitor (C)
