A Capacitor on an AC Supply

Q. A 60 μF capacitor is connected to a 110 V, 60 Hz ac supply. What is the rms value of the current in the circuit, in amperes? expects 2.49 A

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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?

The playground

Inputs

capacitance

C

60 μF

supply

V_{C}

110 V

frequency

ν

60 Hz

Outputs

current

I

The solution

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 Ruby

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

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

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

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)