Relationship power and amplitude

ImAnEngineer

My book states:

"Power is proportional to amplitude squared"

How can this be derived?


Thank you!

Bob S

One example of power vs. amplitude is electricity. We often refer to voltage or current as the amount of electricity in a load, like a resistor (R ohms). In reality, the power of the electricity in a resistor is the current-squared times R (watts), or the voltage-squared divided by R (watts). We alo talk about the product of voltage and current as power. So current or voltage are amplitudes, and the power is watts.

This is also true in electromagnetic waves. We talk about the transverse E vector or the H vector (amplitudes) in an electromagnetic wave. [Note that E is volts per meter and H is amp-turns per meter]. E and H are amplitudes, while the Poynting vector, which is E x H , is power (per square meter).

Added note: The impedance Z of free space is about 377 ohms, so the power of an EM wave can also be written as E²/Z watts per m², or H²Z watts per m².

Amrit N

i think we have learnt in the sound chapter that intensity is proportional to the amplitude.you know Intensity is more or less can be assumed to be power. power per unit area ok . we all know we can derive this by disintigrating wave equation. the plot of Voltage vs time or current vs time also give a wave diagram in graph, we can obtain wave equation from this and after certain procedure as similar that we had in sound chapter(that is relating with energy and linking with certain rate of time) that is likely to give intensity and amplitude relaton, well i am just studying 10+2 class and this is engineering topic but i have well trie to explain this

ImAnEngineer

I agree.
I think this goes way over my head. Never heard of Poynting vectors or transverse vectors.

Can't make a lot of sense of this either.


Thanks for the posts guys. I'll probably have to wait to when I get to the point I can understand this. If there's a more elementary way of explaining this I'd be glad to find out :).

Bob S

John Henry Poynting (1852-1914), one of Maxwell's students, was a professor of physics at Mason Science College, now the University of Birmingham, England. In 1884-1885 he published papers which showed that energy flow can be expressed in a simple formula using the electric and magnetic fields.

Amrit N

you can make sense now
you know
Amplitude means the highest position of the vibrating particle. in the graph plot of Voltage vs time or Current vs time you get sinosoidal curve, the peak yalue of current(I) is called as amplitude . i guess the amplitude in n ur book means that.look here i am doing calculation
I=Ipeak . sinwt (Ipeak=amplitude)(t=instantaneous time)
now power=I^2 .R=(Ipeaksinwt)^2 .R=Ipeak^2.sinwt^2.R=a^2.sinwt^2.R
(here a=amplitude=peak value of current=Ipeak) and that is power proportional to amplitude square)
hey guy i think u have understood now. i think there is no other clear cut way

ImAnEngineer

I understand the terminology. What I don't understand is why P ~ A². You assume P=I², which I'm not so sure about (how do you derive it?) and I don't know what R means here.


PS: your post would be easier to read if you use the x² and x₂ button in the toolbar, and if you use some spaces in your equations

Amrit N

you know
e=electroic charge,V=volt(potencial difference) when 1 electonic charge is travelled in between two points separated by 1 V potential differense then it is called 1 joule work done

you know joule per second is power
that means P=e/s . V(e=electonic charge ,V=potential diff, s=second)
e/s is charge flowing in unit time called current ,that means I=e/s, that means P=IV

you know V=IR(where R = resistance)
Now substituting in P=IV,we get P=I.IR=I^2.R p porportional to I^2(where I=amplitude)
and remaining calculation i have alearady provided already
i hope u understand

Amrit N

P=IV power is product of the current and potential difference
P=IV=I.IR=I^2.R (because V=IR as per ohms law, here R=resistance)
when resstance value is constant P is proportional to I^2
i am sure the amplitude in ur book is peak value of current.

I=Ipeak.sinwt
P=(Ipeak.sinwt)^2.R =(Ipeak)^2.(sinwt)^2.R
here we can achieve, P proportional to (Ipeak)^2 = p proportional to (amplitude)^2


this is reeality ,i guess u understand now

Chronos

Amrit is fundamentally correct. The language barrier is irrelevant when the math is correct, and it is.

Topher925

It may be correct but its a very poor definition.

The power of a signal or waveform is the signal or waveform multiplied by its conjugate.

Power = U x U*, where U is the function describing the signal.

For example;

U = Ae^(t/b)

Where U is the signal function, A is the amplitude, t is time (or what ever you want it to be), and b is just a coefficient.

Since P = U x U*

P = Ae^(t/b) x Ae^(-t/b) = A²

Amrit N

hey topher 92 donot flaoat in maths (iamengineer)said power proportional to amplitue square and i donot understand power = A^2, i am high school student , i have not never read bachelor physics,i suppose u r bachelor , however using common sense i can tell that the amplitude that iamanengineer mentioned is the highest peak of current signal, and surely he didnot mean the amplitude of the particles because there are no particles vibrating in electricity as in sound propagation, the current signal VS time provide the concept of fluctuating current and there is the amplitude,i request topher925 be more physical then mathematical

Topher925

If you don't fundamentally understand the question, then don't try to answer it.

No you can't, you just made the assumption.

Yes there is, look up AC electrical power. I'm guessing the electrons powering your computer are vibrating back and forth 50 times a second.

You want me to answer a question asking how something is mathematically derived without using mathematics?

Amrit, the bottom line is that your definition is only true for some simple cases, but not all. Do you understand why?

Pop Quiz:
(You can solve this with simple HS math and the definition I gave, but try yours)

Lets assume that the AC electricity waveform that is powering your computer can be described with the equation;

U = 5 cos(wt) + 10 isin(wt)

What is the power of this electrical waveform?

Amrit N

U = 5 cos(wt) + 10 isin(wt)
hey tell me what is is U what is i, waht are u trying tell ,that the calculation i provide is the topic from my book, u r trying to tell i am wrong or my concept or my book was wrong , we are here only here to discuss and help others with own assumption that may or may not be perfect
Was Newton faiure for not giving correct form for relative velocity than albert eienstine,In newtons time there were no sufficient evidences of astronomical experiences and findings ,so he failed , that does not mean newton was less brilliant, be more conceptual than just finding actual math.No theories wer made initially complex they were made complex after the thread was gaine by someone , here U that u supposed which is function of some kind of sighnal has some relationship aith peak value of current, if u doubt please carefully do the research
At lasy i wanna ask u what level student are u
the U is not probably in high school physics , i never find it

Amrit N

Yes there is, look up AC electrical power. I'm guessing the electrons powering your computer are vibrating back and forth 50 times a second
and one thing i like to say that my definition tells tthat one u mentioned this , the electrons moving back and forth is indicated in wave diagram graph of Current vs time
u know the vibrating electons forth representd as ny upper half cycle and vibrating back represente my lower half cycle and amplitude means the highrst number of electrons flowing per unit second and then remaining calculation u will see below--------------------------------------------------------------------------------

P=IV power is product of the current and potential difference
P=IV=I.IR=I^2.R (because V=IR as per ohms law, here R=resistance)
when resstance value is constant P is proportional to I^2
i am sure the amplitude in ur book is peak value of current.

I=Ipeak.sinwt
P=(Ipeak.sinwt)^2.R =(Ipeak)^2.(sinwt)^2.R
here we can achieve, P proportional to (Ipeak)^2 = p proportional to (amplitude)^2


this is reeality ,i guess u understand now
u cannot say others wrong if u have learnt things in more special way, look the concept

Amrit N

p= (5coswt+10i sinwt) (5coswt-10isinwt)=(5coswt)^2-(10isinwt)^2= 25cos^2wt+100(sinwt)^2=25(1-sinwt^2)+100sinwt^2=25-75sinwt^2

Amrit N

Now i understand that U is function osf ac signal i is imaginary number i is root under -1
this U , i guess it will be in form (I, V) that is (I+iV) or (V, I) that is (V+iI),, this coordinate which can be obtained by plooting graph of Current ,Voltage Vs time and since one coordinate is in form of coswt and othere is sinwt the current and voltafe have phase differnce of 90 degrees
This all things same
that means u and i are talking same thing
only language is different , mind it ok
pls response topher925