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 learned 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]

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.

I agree.

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).

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

i think we have learned 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

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]

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

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]

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..

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]

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

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 traveled 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]

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

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 don't flaoat in maths (iamengineer)said power proportional to amplitue square and i don't 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]

hey topher 92 don't flaoat in maths (iamengineer)said power proportional to amplitue square and i don't understand power = A^2

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

, however using common sense i can tell that the amplitude that iamanengineer mentioned is the highest peak of current signal

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

, and surely he didnot mean the amplitude of the particles because there are no particles vibrating in electricity as in sound propagation,

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.

i request topher925 be more physical then mathematical

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, what 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 want to 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 learned 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

 

[Amrit N]

you gave U=(5coswt+i10sinwt)=(5coswt,10sinwt), its coordinate obtained by graph of Voltage vs current , that means (V,I) anyway ac signal u told comprises of V and I
any way the power is the product of current and voltage which is included in ur definition of U
you defined amplitude as the length of the displacement of electron from mean position but i defined it in the form (I vs time graph)that means from sinosuidal curve, anway both can support for amplitude at equal way i meanst to say I^2 proportional to A^2 proportional to P

 

[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

you gave U=(5coswt+i10sinwt)=(5coswt,10sinwt), its coordinate obtained by graph of Voltage vs current , that means (V,I) anyway ac signal u told comprises of V and I
any way the power is the product of current and voltage which is included in ur definition of U
you defined amplitude as the length of the displacement of electron from mean position but i defined it in the form (I vs time graph)that means from sinosuidal curve, anway both can support for amplitude at equal way i meanst to say I^2 proportional to A^2 proportional to P

 

[Amrit N]

If you don't fundamentally understand the question, then don't try to answer it.
I am not student of bachelor but i lack terminologies but not conscience


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

Assumption is the key to science(physics)

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.
i mentioned the evidence of the graph plot of the current vs time, the upper half cycle of the ac current is forth vibration of the electron and another half cycle is of back vibration of electron and calculation are all based on it


You want me to answer a question asking how something is mathematically derived without using mathematics?
are u gone out of ur mind what are u telling

Amrit, the bottom line is that your definition is only true for some simple cases, but not all. Do you understand why?
Concept is main and it applies to every complex matter
Pop Quiz:

(You can solve this with simple HS math and the definition I gave, but try yours)
i neednot solve this, i am not here to give test , i am here to help others with my hypothesis

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?

, this is hell

you described the amplitude as the dispacement length of electron about its mean position but i thought peak current as the amplitude
u know P proportional to Ipeak square which is proportional to amplitude square
u know its different approach

 

[Amrit N]

hey u mentioned that the the electrons vibrate back and forth ,this is in my evidence also I=Ipeak sinwt(o,pie,2pie) the current moves in forward direction in phase 0 to pie , that means the electrons vibrate in front and the current flow in opposite direction from phase pie to 2pie ,that means electrons vibrate in backward
hey now i have analyzed ur definition is by the length of displacement of electron in its mean position and i talked abt the maximum current in any half cycle of a.c
the approach may be different but saying one is to indicate the happening of others
mean to say at the peak value of the current the electron achieves the highest vibrating position which is amplitude
I have drawn conclusion that the
Power propotional to (Ipeak)^2 which is proportional to (Amplitude)^2
is that ok

 

[Amrit N]

Any way the power is the rate of doing work to achieve maximum vibration of the partcles .which can be sensed with both peak value of ac and amplitude of electron.ok