Chapter 8 Capacitance

Question 1

Defining Capacitance

Answer 1

• 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

Question 2

• Capacitors are marked with a value of their capacitance. This is defined as:

Answer 2

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

Question 3

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

Answer 3

Question 4

Calculating Capacitance

Answer 4

• 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)

Question 5

If the capacitor is made of parallel plates

Answer 5

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

Question 6

Capacitance of a Spherical Conductor

Answer 6

• 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

Question 7

Capacitors in Series

Answer 7

• 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/C_total

• Substituting these into the equation V = V₁ + V₂ equals:

Q/C_total=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/C_total= 1/C₁ + 1C₂ …

Question 8

Capacitors in Parallel

Answer 8

• 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 = C_totalV

• Substituting these into the Q = Q₁ + Q₂ equals:

C_totalV = 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:

C_total = C₁ + C₂ + C₃ …

Question 9

Capacitors connected in parallel have the same p.d across them, but different charge

Answer 9

Question 10

Capacitors in Series & Parallel

Answer 10

• Recall the formula for the combined capacitance of capacitors In parallel:

C_total = C₁ + C₂ + C₃ …

• in series:

Question 11

Area Under a Potential–Charge Graph

Answer 11

• 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

Question 12

greater amount of work must be done to increase the charge on the negative plate or in other words:

Answer 12

The potential difference V across the capacitor increases as the amount of charge Q increases

Question 13

As the charge on the negative plate builds up, more work needs to be done to add more charge

Answer 13

Question 14

The electric potential energy stored in the capacitor is the area under the potential-charge graph

Answer 14

• 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

Question 15

Calculating Energy Stored in a Capacitor

Answer 15

• 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

Question 16

Capacitors are discharged through a

Answer 16

• resistor
  
  • The electrons now flow back from the negative plate to the positive plate until there are equal numbers on each plate
• At the start of discharge, the current is large (but in the opposite direction to when it was charging) and gradually falls to zero

Question 17

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

Answer 17

Question 18

As a capacitor discharges, the current, p.d and charge all … what?

Answer 18

• decrease exponentially
• The 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 represent an exponential decay

Question 19

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

Answer 19

Question 20

The key features of the discharge graphs are:

Answer 20

• • 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 value starts on the y axis and decreases exponentially
• The rate at which a capacitor discharges depends on the resistance of the circuit
  
  • If the resistance is high, the current will decrease 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 increase and charge will flow from the capacitor plates quickly, meaning the capacitor will discharge faster

Question 21

The Time Constant

Answer 21

• 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 is:

The time taken for the charge of a capacitor to decrease to 0.37 of its original value

• This is represented by the greek letter tau (τ) and measured in units of seconds (s)
• The time constant gives an easy way to compare the rate of change of similar quantities eg. charge, current and p.d.

Question 22

• The time constant is defined by the equation:

Answer 22

τ = RC

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

Question 23

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

Answer 23

Question 24

Using the Capacitor Discharge Equation

Answer 24

• 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)

Question 25

This equation shows what?

Answer 25

* 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

Question 26

Answer 26

* Where: * V = p.d across the capacitor (C) * V₀ = initial p.d across the capacitor (C)

Question 27

The Exponential Function *e*

Answer 27

* The symbol e represents the exponential constant, a number which is approximately equal to e = 2.718… * On a calculator it is shown by the button e^x * The inverse function of e^x is ln(y), known as the natural logarithmic function * This is because, if e^x = y, then x = ln (y) * The 0.37 in the definition of the **time constant** arises as a result of the exponential constant, the true definition is: * The time taken for the charge of a capacitor to decrease by 1/e of its original value * Where 1/e = 0.3678..