Three Capacitors in Parallel

Q. Three capacitors of capacitances 2 pF, 3 pF and 4 pF are connected in parallel. What is the total capacitance of the combination, in pF? expects 9 pF

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Q. Three capacitors of capacitances 2 pF $C_{f} = 2$ pF, 3 pF $C_{s} = 3$ pF and 4 pF $C_{t} = 4$ pF are connected in parallel $C_{c} = C_{f} + C_{s} + C_{t}$ . What is the total capacitance of the combination, in pF?

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Inputs

first

C_{f}

2 pF

second

C_{s}

3 pF

third

C_{t}

4 pF

Outputs

total

C_{c}

The solution

Solved with Capacitance

Given

$C_{f} = 2$ pF
$C_{s} = 3$ pF
$C_{t} = 4$ pF
$C_{c} = C_{f} + C_{s} + C_{t}$ from “connected in parallel”

Find $C_{c}$ .

Working

$C_{f} = 2$ pF $= 2 \times 10^{−12}$ F
$C_{s} = 3$ pF $= 3 \times 10^{−12}$ F
$C_{t} = 4$ pF $= 4 \times 10^{−12}$ F
connected in parallel $C_{c} = C_{f} + C_{s} + C_{t}$
$C_{c} = 2 \times 10^{−12} + 3 \times 10^{−12} + 4 \times 10^{−12}$
$C_{c} = 9$ pF
$C_{c} = 9 \times 10^{−12}$ F $= 9$ pF
$C_{c} \approx 9$ pFto 1 significant figure

Conclusion

Side by side, the capacitors each hold charge at the same voltage, so their capacitances simply add.

The Ruby

1require_relative "../physics"23question :three_capacitors_in_parallel do4  kind :numerical5  describes "Three capacitors of capacitances %{first}, %{second} and %{third} are connected in parallel. " \6            "%{total}, in pF?"7  given first: "2 pF"8  given second: "3 pF"9  given third: "4 pF"10  asks total: "What is the total capacitance of the combination"11end

1require_relative "../questions/three_capacitors_in_parallel"2require_relative "../science/electrostatic_potential_and_capacitance/capacitance"34solve :three_capacitors_in_parallel, using: Capacitance do5  objects :first, :second, :third, :combination67  take first: first.capacitance8  take second: second.capacitance9  take third: third.capacitance1011  deduce_from "connected in parallel",12              combination.capacitance => first.capacitance + second.capacitance + third.capacitance1314  total = combination.capacitance15  answer total:, in: "pF", figures: 11617  conclusion "Side by side, the capacitors each hold charge at the same voltage, so their capacitances simply add."18end

1require_relative "../solutions/three_capacitors_in_parallel"23play_with :three_capacitors_in_parallel do4  title "Three Capacitors 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 first	$Q_{f}$	C	—
capacitance of first	$C_{f}$	F	0 … ∞
voltage of first	$V_{f}$	V	—
charge of second	$Q_{s}$	C	—
capacitance of second	$C_{s}$	F	0 … ∞
voltage of second	$V_{s}$	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 first	$K_{f}$	—	—
area of first	$A_{f}$	m²	—
plate separation of first	$d_{f}$	m	—
dielectric constant of second	$K_{s}$	—	—
area of second	$A_{s}$	m²	—
plate separation of second	$d_{s}$	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 first	$U_{f}$	J	—
energy stored of second	$U_{s}$	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_{f} = C_{f} \cdot V_{f}

charge on a capacitor(first)

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

charge on a capacitor(second)

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_{f} = \frac{K_{f} \cdot ε_{0} \cdot A_{f}}{d_{f}}

a parallel plate capacitor(first)

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

a parallel plate capacitor(second)

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_{f} = \frac{C_{f} \cdot {V_{f}}^{2}}{2}

energy in a capacitor(first)

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

energy in a capacitor(second)

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)

C_{c} = C_{f} + C_{s} + C_{t}

connected in parallel