The Charge on a Capacitor in Parallel

Q. Three capacitors of capacitances 2 pF, 3 pF and 4 pF are connected in parallel, and the combination is connected to a 100 V supply. Determine the charge on the last of them, in coulombs? expects 4 × 10⁻¹⁰ C

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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?

The playground

Inputs

third

C_{t}

4 pF

supply

V_{c}

100 V

Outputs

charge

Q_{t}

The solution

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 Ruby

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

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

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

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