The Charge on a Capacitor in Parallel

solved Capacitance

Q. Three capacitors of capacitances 2 pF, 3 pF and 4 pF $C_{t} = 4$ pF are connected in parallel $V_{t} = V_{c}$ , and the combination is connected to a 100 V $V_{c} = 100$ V supply. Determine the charge on the last of them, in coulombs?

Inputs

third

C_{t}

4 pF

supply

V_{c}

100 V

Outputs

charge

Q_{t}

How it's solved

Solved with Capacitance

Given

$C_{t} = 4$ pF
$V_{c} = 100$ V
$V_{t} = V_{c}$ from “connected in parallel”

Find $Q_{t}$ .

Working

$C_{t} = 4$ pF $= 4 \times 10^{−12}$ F
connected in parallel $V_{t} = V_{c}$
$V_{t} = 100$
$V_{t} = 100$ V
charge on a capacitor(third) $Q_{t} = C_{t} \cdot V_{t}$
$Q_{t} = 4 \times 10^{−12} \cdot 100$
$Q_{t} = 4 \times 10^{−10}$ C
$Q_{t} \approx 4 \times 10^{−10}$ Cto 1 significant figure

Conclusion

In parallel each capacitor has the whole supply voltage across it, so each holds its own capacitance times that voltage.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :the_charge_on_a_capacitor_in_parallel do4  kind :numerical5  describes "Three capacitors of capacitances 2 pF, 3 pF and %{third} are connected in parallel, and the " \6            "combination is connected to a %{supply} supply. %{charge}, in coulombs?"7  given third: "4 pF"8  given supply: "100 V"9  asks charge: "Determine the charge on the last of them"10end

solution.rb

1require_relative "../questions/the_charge_on_a_capacitor_in_parallel"2require_relative "../science/electrostatic_potential_and_capacitance/capacitance"34solve :the_charge_on_a_capacitor_in_parallel, using: Capacitance do5  objects :third, :combination67  take third: third.capacitance8  take supply: combination.voltage910  deduce_from "connected in parallel", third.voltage => combination.voltage1112  charge = third.charge13  answer charge:, in: "C", figures: 11415  conclusion "In parallel each capacitor has the whole supply voltage across it, so each holds its own " \16             "capacitance times that voltage."17end

playground.rb

1require_relative "../solutions/the_charge_on_a_capacitor_in_parallel"23play_with :the_charge_on_a_capacitor_in_parallel do4  title "The Charge on a Capacitor in Parallel"5end

The laws it drew on

quantity	written	units	within	from
permittivity of free space	$ε_{0}$	F/m	—	Capacitance
capacitance	$C$	F	0 … ∞	Capacitance
charge	$Q$	C	—	Capacitance
voltage	$V$	V	—	Capacitance
energy stored	$U$	J	—	Capacitance
area	$A$	m²	—	Capacitance
plate separation	$d$	m	—	Capacitance
dielectric constant	$K$	—	—	Capacitance
charge of ground	$Q_{g}$	C	—
capacitance of ground	$C_{g}$	F	0 … ∞
voltage of ground	$V_{g}$	V	—
charge of third	$Q_{t}$	C	—
capacitance of third	$C_{t}$	F	0 … ∞
voltage of third	$V_{t}$	V	—
charge of combination	$Q_{c}$	C	—
capacitance of combination	$C_{c}$	F	0 … ∞
voltage of combination	$V_{c}$	V	—
dielectric constant of ground	$K_{g}$	—	—
area of ground	$A_{g}$	m²	—
plate separation of ground	$d_{g}$	m	—
dielectric constant of third	$K_{t}$	—	—
area of third	$A_{t}$	m²	—
plate separation of third	$d_{t}$	m	—
dielectric constant of combination	$K_{c}$	—	—
area of combination	$A_{c}$	m²	—
plate separation of combination	$d_{c}$	m	—
energy stored of ground	$U_{g}$	J	—
energy stored of third	$U_{t}$	J	—
energy stored of combination	$U_{c}$	J	—

Q_{c} = C_{c} \cdot V_{c}

charge on a capacitor

C_{c} = \frac{K_{c} \cdot ε_{0} \cdot A_{c}}{d_{c}}

a parallel plate capacitor

U_{c} = \frac{C_{c} \cdot {V_{c}}^{2}}{2}

energy in a capacitor

Q_{g} = C_{g} \cdot V_{g}

charge on a capacitor(ground)

Q_{t} = C_{t} \cdot V_{t}

charge on a capacitor(third)

Q_{c} = C_{c} \cdot V_{c}

charge on a capacitor(combination)

C_{g} = \frac{K_{g} \cdot ε_{0} \cdot A_{g}}{d_{g}}

a parallel plate capacitor(ground)

C_{t} = \frac{K_{t} \cdot ε_{0} \cdot A_{t}}{d_{t}}

a parallel plate capacitor(third)

C_{c} = \frac{K_{c} \cdot ε_{0} \cdot A_{c}}{d_{c}}

a parallel plate capacitor(combination)

U_{g} = \frac{C_{g} \cdot {V_{g}}^{2}}{2}

energy in a capacitor(ground)

U_{t} = \frac{C_{t} \cdot {V_{t}}^{2}}{2}

energy in a capacitor(third)

U_{c} = \frac{C_{c} \cdot {V_{c}}^{2}}{2}

energy in a capacitor(combination)

V_{t} = V_{c}

connected in parallel