# 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.

Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
hi
Loga
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 Kimberly Nichols By OpenStax By OpenStax By Madison Christian By Brianna Beck By Yasser Ibrahim By Mary Matera By Brooke Delaney By David Geltner By