<< Chapter < Page Chapter >> Page >

Note that although the resistance in the circuit considered is negligible, the AC current is not extremely large because inductive reactance impedes its flow. With AC, there is no time for the current to become extremely large.

Capacitors and capacitive reactance

Consider the capacitor connected directly to an AC voltage source as shown in [link] . The resistance of a circuit like this can be made so small that it has a negligible effect compared with the capacitor, and so we can assume negligible resistance. Voltage across the capacitor and current are graphed as functions of time in the figure.

Part a of the figure shows a capacitor C connected across an A C voltage source V. The voltage across the capacitor is given by V C. Part b of the diagram shows a graph for the variation of current and voltage across the capacitor as functions of time. The voltage V C and current I C is plotted along the Y axis and the time t is along the X axis. The graph for current is a progressive sine wave from the origin starting with a wave along the negative Y axis. The graph for voltage is a cosine wave and amplitude slightly less than the current wave.
(a) An AC voltage source in series with a capacitor C having negligible resistance. (b) Graph of current and voltage across the capacitor as functions of time.

The graph in [link] starts with voltage across the capacitor at a maximum. The current is zero at this point, because the capacitor is fully charged and halts the flow. Then voltage drops and the current becomes negative as the capacitor discharges. At point a, the capacitor has fully discharged ( Q = 0 size 12{Q=0} {} on it) and the voltage across it is zero. The current remains negative between points a and b, causing the voltage on the capacitor to reverse. This is complete at point b, where the current is zero and the voltage has its most negative value. The current becomes positive after point b, neutralizing the charge on the capacitor and bringing the voltage to zero at point c, which allows the current to reach its maximum. Between points c and d, the current drops to zero as the voltage rises to its peak, and the process starts to repeat. Throughout the cycle, the voltage follows what the current is doing by one-fourth of a cycle:

Ac voltage in a capacitor

When a sinusoidal voltage is applied to a capacitor, the voltage follows the current by one-fourth of a cycle, or by a 90º phase angle.

The capacitor is affecting the current, having the ability to stop it altogether when fully charged. Since an AC voltage is applied, there is an rms current, but it is limited by the capacitor. This is considered to be an effective resistance of the capacitor to AC, and so the rms current I size 12{I} {} in the circuit containing only a capacitor C size 12{C} {} is given by another version of Ohm’s law to be

I = V X C , size 12{I= { {V} over {X rSub { size 8{C} } } } } {}

where V size 12{V} {} is the rms voltage and X C size 12{X rSub { size 8{C} } } {} is defined (As with X L size 12{X rSub { size 8{L} } } {} , this expression for X C size 12{X rSub { size 8{C} } } {} results from an analysis of the circuit using Kirchhoff’s rules and calculus) to be

X C = 1 fC , size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } } {}

where X C size 12{X rSub { size 8{C} } } {} is called the capacitive reactance    , because the capacitor reacts to impede the current. X C size 12{X rSub { size 8{C} } } {} has units of ohms (verification left as an exercise for the reader). X C size 12{X rSub { size 8{C} } } {} is inversely proportional to the capacitance C size 12{C} {} ; the larger the capacitor, the greater the charge it can store and the greater the current that can flow. It is also inversely proportional to the frequency f size 12{f} {} ; the greater the frequency, the less time there is to fully charge the capacitor, and so it impedes current less.

Calculating capacitive reactance and then current

(a) Calculate the capacitive reactance of a 5.00 mF capacitor when 60.0 Hz and 10.0 kHz AC voltages are applied. (b) What is the rms current if the applied rms voltage is 120 V?

Strategy

The capacitive reactance is found directly from the expression in X C = 1 fC size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } } {} . Once X C has been found at each frequency, Ohm’s law stated as I = V / X C size 12{I=V/X rSub { size 8{C} } } {} can be used to find the current at each frequency.

Solution for (a)

Entering the frequency and capacitance into X C = 1 fC size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } } {} gives

X C = 1 fC = 1 6 . 28 ( 60 . 0 / s ) ( 5 . 00  μ F ) = 531 Ω at 60 Hz . alignl { stack { size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } } {} #" "= { {1} over {6 "." "28" \( "60" "." 0/s \) \( 5 "." "00" μF \) } } ="531 " %OMEGA " at 60 Hz" {} } } {}

Similarly, at 10 kHz,

X C = 1 fC = 1 6 . 28 ( 1 . 00 × 10 4 / s ) ( 5 . 00  μ F ) = 3.18 Ω at 10 kHz . alignl { stack { size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } = { {1} over {6 "." "28" \( 1 "." "00" times "10" rSup { size 8{4} } /s \) \( 5 "." "00" μF \) } } } {} #" "=3 "." "18" %OMEGA " at 10 kHz" {} } } {}

Solution for (b)

The rms current is now found using the version of Ohm’s law in I = V / X C size 12{I=V/X rSub { size 8{C} } } {} , given the applied rms voltage is 120 V. For the first frequency, this yields

I = V X C = 120 V 531 Ω = 0.226 A at 60 Hz . size 12{I= { {V} over {X rSub { size 8{C} } } } = { {"120"" V"} over {"531 " %OMEGA } } =0 "." "226"" A"} {}

Similarly, at 10 kHz,

I = V X C = 120 V 3.18 Ω = 37.7 A at 10 kHz . size 12{I= { {V} over {X rSub { size 8{C} } } } = { {"120"" V"} over {3 "." "18 " %OMEGA } } ="37" "." 7" A"} {}

Discussion

The capacitor reacts very differently at the two different frequencies, and in exactly the opposite way an inductor reacts. At the higher frequency, its reactance is small and the current is large. Capacitors favor change, whereas inductors oppose change. Capacitors impede low frequencies the most, since low frequency allows them time to become charged and stop the current. Capacitors can be used to filter out low frequencies. For example, a capacitor in series with a sound reproduction system rids it of the 60 Hz hum.

Got questions? Get instant answers now!

Questions & Answers

Why is the sky blue...?
Star Reply
It's filtered light from the 2 forms of radiation emitted from the sun. It's mainly filtered UV rays. There's a theory titled Scatter Theory that covers this topic
Mike
A heating coil of resistance 30π is connected to a 240v supply for 5min to boil a quantity of water in a vessel of heat capacity 200jk. If the initial temperature of water is 20°c and it specific heat capacity is 4200jkgk calculate the mass of water in a vessel
fasawe Reply
A thin equi convex lens is placed on a horizontal plane mirror and a pin held 20 cm vertically above the lens concise in position with its own image the space between the undersurface of d lens and the mirror is filled with water (refractive index =1•33)and then to concise with d image d pin has to
Azummiri Reply
Be raised until its distance from d lens is 27cm find d radius of curvature
Azummiri
what happens when a nuclear bomb and atom bomb bomb explode add the same time near each other
FlAsH Reply
A monkey throws a coconut straight upwards from a coconut tree with a velocity of 10 ms-1. The coconut tree is 30 m high. Calculate the maximum height of the coconut from the top of the coconut tree? Can someone answer my question
Fatinizzah Reply
v2 =u2 - 2gh 02 =10x10 - 2x9.8xh h = 100 ÷ 19.6 answer = 30 - h.
Ramonyai
why is the north side is always referring to n side of magnetic
sam Reply
who is a nurse
Chilekwa Reply
A nurse is a person who takes care of the sick
Bukola
a nurse is also like an assistant to the doctor
Gadjawa
explain me wheatstone bridge
Malik Reply
good app
samuel
Wheatstone bridge is an instrument used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which includes the unknown component.
MUHD
Rockwell Software is Rockwell Automation’s "Retro Encabulator". Now, basically the only new principle involved is that instead of power being generated by the relative motion of conductors and fluxes, it’s produced by the modial interaction of magneto-reluctance and capacitive diractance. The origin
Chip
what refractive index
Adjah Reply
write a comprehensive note on primary colours
Harrison Reply
relationship between refractive index, angle of minimum deviation and angle of prism
Harrison
Who knows the formula for binding energy,and what each variable or notation stands for?
Agina Reply
1. A black thermocouple measures the temperature in the chamber with black walls.if the air around the thermocouple is 200 C,the walls are at 1000 C,and the heat transfer constant is 15.compute the temperature gradient
Tikiso Reply
what is the relationship between G and g
Olaiya Reply
G is the u. constant, as g stands for grav, accelerate at a discreet point
Mark
Is that all about it?
Olaiya
pls explain in details
Olaiya
G is a universal constant
Mark
g stands for the gravitational acceleration point. hope this helps you.
Mark
balloon TD is at a gravitational acceleration at a specific point
Mark
I'm sorry this doesn't take dictation very well.
Mark
Can anyone explain the Hooke's law of elasticity?
Olaiya Reply
extension of a spring is proportional to the force applied so long as the force applied does not exceed the springs capacity according to my textbook
Amber
does this help?
Amber
Yes, thanks
Olaiya
so any solid can be compressed how compressed is dependent upon how much force is applied F=deltaL
Amber
sorry, the equation is F=KdeltaL delta is the triangle symbol and L is length so the change in length is proportional to amount of Force applied I believe that is what Hookes law means. anyone catch any mistakes here please correct me :)
Amber
I think it is used only for solids and not liquids, isn't it?
Olaiya
basically as long as you dont exceed the elastic limit the object should return to it original form but if you exceed this limit the object will not return to original shape as it will break
Amber
Thanks for the explanation
Olaiya
yh, liquids don't apply here, that should be viscosity
Chiamaka
hope it helps 😅
Amber
also, an object doesnt have to break necessarily, but it will have a new form :)
Amber
Yes
Olaiya
yeah, I think it is for solids but maybe there is a variation for liquids? that I am not sure of
Amber
ok
Olaiya
good luck!
Amber
Same
Olaiya
aplease i need a help on spcific latent heat of vibrations
Bilgate
specific latent heat of vaporisation
Bilgate
how many kilometers makes a mile
Margaret Reply
about 1.6 kilometres.
Faizyab
near about 1.67 kilometers
Aakash
equal to 1.609344 kilometers.
MUHD
Practice Key Terms 2

Get the best College physics course in your pocket!





Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask