A Capacitor on an AC Supply

solved Reactance

Q. A 60 μF $C = 60$ μF capacitor is connected to a 110 V $V_{C} = 110$ V, 60 Hz $ν = 60$ Hz ac supply. What is the rms value of the current in the circuit, in amperes?

Inputs

capacitance

C

60 μF

supply

V_{C}

110 V

frequency

ν

60 Hz

Outputs

current

I

How it's solved

Solved with Reactance

Given

$C = 60$ μF
$V_{C} = 110$ V
$ν = 60$ Hz

Find $I$ .

Working

$C = 60$ μF $= 6 \times 10^{−5}$ F
reactance of a capacitor(circuit) $X_{C} = \frac{1}{2 \cdot π \cdot ν \cdot C}$
$X_{C} = \frac{1}{2 \cdot 3.14159265 \cdot 60 \cdot 6 \times 10^{−5}}$
$X_{C} = 44.2097064$ Ω
current through the capacitor(circuit) $V_{C} = I \cdot X_{C}$
$110 = I \cdot 44.2097064$
$I = 2.488$ A
$I \approx 2.49$ Ato 3 significant figures

Conclusion

The capacitor alone opposes the current, by a reactance that shrinks as the frequency and the capacitance grow.

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 :a_capacitor_on_an_ac_supply do4  kind :numerical5  describes "A %{capacitance} capacitor is connected to a %{supply}, %{frequency} ac supply. %{current}, in " \6            "amperes?"7  given capacitance: "60 μF"8  given supply: "110 V"9  given frequency: "60 Hz"10  asks current: "What is the rms value of the current in the circuit"11end

solution.rb

1require_relative "../questions/a_capacitor_on_an_ac_supply"2require_relative "../science/alternating_current/reactance"34solve :a_capacitor_on_an_ac_supply, using: Reactance do5  objects :circuit67  take capacitance: circuit.capacitance8  take supply: circuit.voltage_across_the_capacitor9  take frequency: circuit.frequency1011  current = circuit.rms_current12  answer current:, in: "A", figures: 31314  conclusion "The capacitor alone opposes the current, by a reactance that shrinks as the frequency and the " \15             "capacitance grow."16end

playground.rb

1require_relative "../solutions/a_capacitor_on_an_ac_supply"23play_with :a_capacitor_on_an_ac_supply do4  title "A Capacitor on an AC Supply"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	Reactance
rms voltage	$V$	V	0 … ∞	PeakAndRms
peak voltage	$v_{m}$	V	—	PeakAndRms
rms current	$I$	A	0 … ∞	PeakAndRms
peak current	$i_{m}$	A	—	PeakAndRms
frequency	$ν$	Hz	0 … ∞	Reactance
angular frequency	$ω$	rad/s	—	Reactance
inductance	$L$	H	—	Reactance
capacitance	$C$	F	0 … ∞	Reactance
inductive reactance	$X_{L}$	Ω	0 … ∞	Reactance
capacitive reactance	$X_{C}$	Ω	0 … ∞	Reactance
voltage across the inductor	$V_{L}$	V	—	Reactance
voltage across the capacitor	$V_{C}$	V	—	Reactance
peak voltage of ground	$v_{m,g}$	V	—
rms voltage of ground	$V_{g}$	V	0 … ∞
peak voltage of circuit	$v_{m}$	V	—
rms voltage of circuit	$V$	V	0 … ∞
peak current of ground	$i_{m,g}$	A	—
rms current of ground	$I_{g}$	A	0 … ∞
peak current of circuit	$i_{m}$	A	—
rms current of circuit	$I$	A	0 … ∞
angular frequency of ground	$ω_{g}$	rad/s	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
angular frequency of circuit	$ω$	rad/s	—
frequency of circuit	$ν$	Hz	0 … ∞
inductive reactance of ground	$X_{L,g}$	Ω	0 … ∞
inductance of ground	$L_{g}$	H	—
inductive reactance of circuit	$X_{L}$	Ω	0 … ∞
inductance of circuit	$L$	H	—
capacitive reactance of ground	$X_{C,g}$	Ω	0 … ∞
capacitance of ground	$C_{g}$	F	0 … ∞
capacitive reactance of circuit	$X_{C}$	Ω	0 … ∞
capacitance of circuit	$C$	F	0 … ∞
voltage across the inductor of ground	$V_{L,g}$	V	—
voltage across the inductor of circuit	$V_{L}$	V	—
voltage across the capacitor of ground	$V_{C,g}$	V	—
voltage across the capacitor of circuit	$V_{C}$	V	—

v_{m,c} = \sqrt{2} \cdot V_{c}

peak is root two times rms

i_{m,c} = \sqrt{2} \cdot I_{c}

peak current is root two times rms

ω_{c} = 2 \cdot π \cdot ν_{c}

angular frequency and frequency

X_{L,c} = 2 \cdot π \cdot ν_{c} \cdot L_{c}

reactance of an inductor

X_{C,c} = \frac{1}{2 \cdot π \cdot ν_{c} \cdot C_{c}}

reactance of a capacitor

V_{L,c} = I_{c} \cdot X_{L,c}

current through the inductor

V_{C,c} = I_{c} \cdot X_{C,c}

current through the capacitor

v_{m,g} = \sqrt{2} \cdot V_{g}

peak is root two times rms(ground)

v_{m} = \sqrt{2} \cdot V

peak is root two times rms(circuit)

i_{m,g} = \sqrt{2} \cdot I_{g}

peak current is root two times rms(ground)

i_{m} = \sqrt{2} \cdot I

peak current is root two times rms(circuit)

ω_{g} = 2 \cdot π \cdot ν_{g}

angular frequency and frequency(ground)

ω = 2 \cdot π \cdot ν

angular frequency and frequency(circuit)

X_{L,g} = 2 \cdot π \cdot ν_{g} \cdot L_{g}

reactance of an inductor(ground)

X_{L} = 2 \cdot π \cdot ν \cdot L

reactance of an inductor(circuit)

X_{C,g} = \frac{1}{2 \cdot π \cdot ν_{g} \cdot C_{g}}

reactance of a capacitor(ground)

X_{C} = \frac{1}{2 \cdot π \cdot ν \cdot C}

reactance of a capacitor(circuit)

V_{L,g} = I_{g} \cdot X_{L,g}

current through the inductor(ground)

V_{L} = I \cdot X_{L}

current through the inductor(circuit)

V_{C,g} = I_{g} \cdot X_{C,g}

current through the capacitor(ground)

V_{C} = I \cdot X_{C}

current through the capacitor(circuit)