CIE A Level Physics复习笔记19.1.2 Derivation of C = Q/V
Derivation of C = Q/V
The circuit symbol for a parallel plate capacitor is two parallel lines
Circuit symbol for a capacitor
Capacitors can be combined in series and parallel circuits
The combined capacitance depends on whether the capacitors are connected in series or parallel
Capacitors in Series
Consider two parallel plate capacitors C1 and C2 connected in series, with a potential difference (p.d) V across them
Capacitors connected in series have different p.d across them but have the same charge
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:
Where the total p.d V is defined by the total capacitance
Substituting these into the equation V = V1 + V2 equals:
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:
Capacitors in Parallel
Consider two parallel plate capacitors C1 and C2 connected in parallel, each with p.d V
Capacitors connected in parallel have the same p.d across them, but different charge
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 ...
Exam Tip
You will be expected to remember these derivations for your exam, therefore, make sure you understand each step. You should especially make sure to revise how the current and potential difference varies in a series and parallel circuit.
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