# 3.5 Alternating current versus direct current  (Page 2/10)

 Page 2 / 10

We are most often concerned with average power rather than its fluctuations—that 60-W light bulb in your desk lamp has an average power consumption of 60 W, for example. As illustrated in [link] , the average power ${P}_{\text{ave}}$ is

${P}_{\text{ave}}=\frac{1}{2}{I}_{0}{V}_{0}.$

This is evident from the graph, since the areas above and below the $\left(1/2\right){I}_{0}{V}_{0}$ line are equal, but it can also be proven using trigonometric identities. Similarly, we define an average or rms current     ${I}_{\text{rms}}$ and average or rms voltage     ${V}_{\text{rms}}$ to be, respectively,

${I}_{\text{rms}}=\frac{{I}_{0}}{\sqrt{2}}$

and

${V}_{\text{rms}}=\frac{{V}_{0}}{\sqrt{2}}.$

where rms stands for root mean square, a particular kind of average. In general, to obtain a root mean square, the particular quantity is squared, its mean (or average) is found, and the square root is taken. This is useful for AC, since the average value is zero. Now,

${P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}},$

which gives

${P}_{\text{ave}}=\frac{{I}_{0}}{\sqrt{2}}\cdot \frac{{V}_{0}}{\sqrt{2}}=\frac{1}{2}{I}_{0}{V}_{0},$

as stated above. It is standard practice to quote ${I}_{\text{rms}}$ , ${V}_{\text{rms}}$ , and ${P}_{\text{ave}}$ rather than the peak values. For example, most household electricity is 120 V AC, which means that ${V}_{\text{rms}}$ is 120 V. The common 10-A circuit breaker will interrupt a sustained ${I}_{\text{rms}}$ greater than 10 A. Your 1.0-kW microwave oven consumes ${P}_{\text{ave}}=\text{1.0 kW}$ , and so on. You can think of these rms and average values as the equivalent DC values for a simple resistive circuit.

To summarize, when dealing with AC, Ohm’s law and the equations for power are completely analogous to those for DC, but rms and average values are used for AC. Thus, for AC, Ohm’s law is written

${I}_{\text{rms}}=\frac{{V}_{\text{rms}}}{R}.$

The various expressions for AC power ${P}_{\text{ave}}$ are

${P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}},$
${P}_{\text{ave}}=\frac{{V}_{\text{rms}}^{2}}{R},$

and

${P}_{\text{ave}}={I}_{\text{rms}}^{2}R.$

## Peak voltage and power for ac

(a) What is the value of the peak voltage for 120-V AC power? (b) What is the peak power consumption rate of a 60.0-W AC light bulb?

Strategy

We are told that ${V}_{\text{rms}}$ is 120 V and ${P}_{\text{ave}}$ is 60.0 W. We can use ${V}_{\text{rms}}=\frac{{V}_{0}}{\sqrt{2}}$ to find the peak voltage, and we can manipulate the definition of power to find the peak power from the given average power.

Solution for (a)

Solving the equation ${V}_{\text{rms}}=\frac{{V}_{0}}{\sqrt{2}}$ for the peak voltage ${V}_{0}$ and substituting the known value for ${V}_{\text{rms}}$ gives

${V}_{0}=\sqrt{2}{V}_{\text{rms}}=\text{1}\text{.}\text{414}\left(\text{120 V}\right)=\text{170 V}.$

Discussion for (a)

This means that the AC voltage swings from 170 V to $\text{–170 V}$ and back 60 times every second. An equivalent DC voltage is a constant 120 V.

Solution for (b)

Peak power is peak current times peak voltage. Thus,

${P}_{0}={I}_{0}{V}_{0}=\text{2}\left(\frac{1}{2}{I}_{0}{V}_{0}\right)=\text{2}{P}_{\text{ave}}.$

We know the average power is 60.0 W, and so

${P}_{0}=\text{2}\left(\text{60}\text{.}\text{0 W}\right)=\text{120 W}.$

Discussion

So the power swings from zero to 120 W one hundred twenty times per second (twice each cycle), and the power averages 60 W.

## Why use ac for power distribution?

Most large power-distribution systems are AC. Moreover, the power is transmitted at much higher voltages than the 120-V AC (240 V in most parts of the world) we use in homes and on the job. Economies of scale make it cheaper to build a few very large electric power-generation plants than to build numerous small ones. This necessitates sending power long distances, and it is obviously important that energy losses en route be minimized. High voltages can be transmitted with much smaller power losses than low voltages, as we shall see. (See [link] .) For safety reasons, the voltage at the user is reduced to familiar values. The crucial factor is that it is much easier to increase and decrease AC voltages than DC, so AC is used in most large power distribution systems.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers! By By  By Rhodes      By Mldelatte