Describe the basic structure of a capacitor.
2x conducting metal plates separated by an insulated dielectric
How do capacitors store electrical energy?
- Work is done to separate electric charges onto opposing plates
- This energy is stored as electric potential energy in the field that forms between the plates
- Hence a p.d.
How do capacitors differ from electrochemical cells?
- They store electrostatic energy rather than chemical energy to be converted to electrical forms
- They can store and release energy far quicker than chemical cells
- Less energy overall can be stored
Describe the general trend of capacitor charging.
- Initially current is very large
- This decays exponentially
- Until p.d. across the oppositely-charged plates rises to match that of the original power supply
- Opposing emf and reducing current
Describe the process of discharging a capacitor.
- They discharge though a resistor with no present power supply; no opposition to V(cap)
- Electrons flow back from the -ve plate to the +ve one until p.d. = 0
- Current is initially large and in the -ve direction
Describe the flow of electrons when charging a capacitor.
- Electron are repelled from the plate closest to the +ve terminal, leaving this plate with a +ve charge
- Electrons are deposited onto the -ve plate leaving it with a net -ve charge
- Until p.d increases to match emf so no current flows
Define capacitance.
- The charge stored per unit potential difference
- This has the units Farad.
Give the equation used to determine capacitance.
Q=CV
How does capacitance in series differ to that in parallel?
In series, total capacitance is given by…
1/C1 + 1/C2 … +1/Cx = 1/Ct
… reflecting the reciprocal rule of resistors IN PARALLEL.
In parallel, total capacitance is given by C1 + C2 … Cx = Ct. This reflects the rules of resistors IN SERIES.
Derive total capacitance in parallel.
Qt = C1V1 + C2V2…
CtVt = C1V1 + C2V2 …
Ct = C1 + C2 … where V is equal across both parallel branches by K2.
Derive capacitance in series.
Vt = V1 + V2…
Qt/Ct = Q1/C1 + Q2/C2 …
Since charge Qt displaced across each capacitor is equal…
1/Ct = 1/C1 + 1/C2 …
Why is work done when we charge a capacitor?
- Separating charges onto an opposing plate forms and intensifies an electric field between the plates
- Work is done to overcome electric force as we remove further electrons
How do we graphically determine the work done by a capacitor?
W=1/2*QV
- This is equal to the area under a V/Q graph
- Remainder is energy dissipated
Derive equations to calculate work done when charging a capacitor.
W=1/2CV^2 = 1/2Q^2/C
Describe the shape of the following graphs for charging a capacitor…
- I/t
- V/t
- I/V is an downwards exponential decay
- V/t is an upwards exponential decay from -ve
Describe the shape of the following graphs for DIScharging a capacitor…
- I/t
- V/t
- I/V is an upwards exponential decay lying within the -ve portion of the I-axis
- V/t is a downwards exponential decay
Explain the trends in current, voltage for charging a capacitor.
As we charge…
- initial current is high
- p.d increases until it matches emf
- current decreases exponentially until reaching zero at this point
Explain the trends in current, voltage for DIScharging a capacitor.
As we discharge…
- initial current is high but -ve
- p.d decreases until it reaches zero
- current decreases exponentially until reaching zero at this point
Give the equation describing charge for a DIScharging capacitor. Derive this for current and voltage.
Q=Q0 * e^-t/CR
I=-I0 * e^-t/CR
V=V0 * e^-t/CR
Give the equation describing charge for a charging capacitor. Derive this for current and voltage.
Q=Q0 * (1 - e^-t/CR)
I=I0 * e^-t/CR
V=V0 * (1 - e^-t/CR)
What is the time constant for the exponential capacitor equations, and how is it calculated?
- Time constant = C * R
- This is the time taken for Q on the capacitor to fall to 1/e of its’ original value.
How do we use exponential modelling to model discharge of a capacitor?
- For a small time interval…
delta Q = (delta T / CR) * Q - We apply this formula iteratively
Why is charging a capacitor an exponential process?
- As the electric field of the capacitor grows, the electric potential between the plates grows
- This acts to oppose the emf so voltage overall decreases
- Thus rate of flow of charge and rate of change of emf, charge decreases over time - Since charge carriers flow less readily
