Three Capacitors in Parallel

solved Capacitance

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?

Inputs

first

C_{f}

2 pF

second

C_{s}

3 pF

third

C_{t}

4 pF

Outputs

total

C_{c}

How it's solved

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 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 :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

solution.rb

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

playground.rb

1require_relative "../solutions/three_capacitors_in_parallel"23play_with :three_capacitors_in_parallel do4  title "Three Capacitors 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 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