What is the integral of energy?

[Boltzman Oscillation]

So the classical law of force given by Newton is F= ma = dp/dt = qE. Thus if i integrate the last two equivalents I get:
∫(dp/dt)dt = q∫Edt
p + C = q∫Edt
correct?
then what would the integral of energy be? I know that E = P/t. I guess I could let P = VI = I^2 * R = (dq/dt)^2 *R eerrr then E = (dq/dt)^2 *R/t and
p + C = qR∫(dq/dt)^2 / t dt
am i getting somewhere or not? I am just curious to see what integrating E can get me in relationship to momentum. Thanks

 

[Dale]

what would the integral of energy be?

The E above is the electric field, not the energy.

 

[Boltzman Oscillation]

The E above is the electric field, not the energy.

im dumb asf bro.

 

[Dale]

im dumb asf bro.

No worries. You just have to keep in mind what the equations mean and not just juggle symbols around.

The integral of E wrt time is the electric impulse per unit charge on a charge experiencing the electric field E. Impulse is the change in momentum, so that matches with the equation you derived.

 

[DrClaude]

Even when combining the latin alphabet with the greek one, there is not enough symbols to go around, and they get reused all the time. Part of learning physics is understanding what each symbol means in a specific context.

 

[Boltzman Oscillation]

No worries. You just have to keep in mind what the equations mean and not just juggle symbols around.

The integral of E wrt time is the electric impulse per unit charge on a charge experiencing the electric field E. Impulse is the change in momentum, so that matches with the equation you derived.

Okay good! That will definitely help me understand what the equation means. I haven't studied electromagnetism in my studies yet so some of these equations are new!

 

[Geoffrey Tullener]

I'm working on battery designs and I wondered the same thing. What do you get when you integrate energy w.r.t. time? Being a lazy engineer with no redeeming value, I googled it and found this post. ... not helpful so I did a thought experiment:

If I integrate Power measured at my battery terminals w.r.t. time and I know my initial conditions - State of Charge (SoC) at t0 - then I can estimate the SoC at any point in time.

If I integrate Energy in my battery (assuming I could measure that) w.r.t time, then I just get a big number. Even at the energy death of the universe (assuming my memory is still working) the number is large but not meaningful.

Can anyone think of a use-case where ∫Edt is useful? ... and if such a thing has a name. Given that jerk snuck up on my in my 40s. Who would have though that the derivative of acceleration had a name? Does mom know?

It wasn't just an academic musing, I was trying to work out if energy was the highest order parameter I need to care about in my energy storage device (dFMEA and P-Diagrams. I think it is but I suspect the physicist in the room are laughing at the engineer who somehow got through the spam filter.

 

[anorlunda]

Let's see. If power has units of watts, then energy has units of watt*hours, and your integral would have units of watt*hours*hours. I don't see where that would be useful.

If we have a factory that makes 1000 batteries per hour, each with X watt*hours capacity, then the factory's output in one day is 24000 X watt*hours*hours. Is that interesting?

 

[mjc123]

There is a quantity called "action" which has the dimensions of energy*time, and a "principle of least action" (https://en.wikipedia.org/wiki/Principle_of_least_action). Planck's constant has the dimensions of action (J s).

 

[Geoffrey Tullener]

Let's see. If power has units of watts, then energy has units of watt*hours, and your integral would have units of watt*hours*hours. I don't see where that would be useful.

If we have a factory that makes 1000 batteries per hour, each with X watt*hours capacity, then the factory's output in one day is 24000 X watt*hours*hours. Is that interesting?

Thank you anorlunda,

I like the dimensional analysis approach. ... embarrassed I didn't go there. In fact, I have seen these figures quoted when expressing the value of the factory to the economy.

So if you wanted to work out how many giga factories you need to support a fleet of electric vehicles ready for 2030, then you would have to how many kwhh you need by reasearching the fleet demand and dividing by the number of kwhh per factory.

I am astounded by how astounded I get whenever I discover another dimension. Have you physicists gotten over it yet?

 

[Geoffrey Tullener]

There is a quantity called "action" which has the dimensions of energy*time, and a "principle of least action" (https://en.wikipedia.org/wiki/Principle_of_least_action). Planck's constant has the dimensions of action (J s).

This is going to take some time to digest. Thanks for the pointer. Now I can't rest until I understand at least 60% of the words in the article and at least 20% of the meaning expressed when you string those words together.

 

[hutchphd]

If we have a factory that makes 1000 batteries per hour, each with X watt*hours capacity, then the factory's output in one day is 24000 X watt*hours*hours. Is that interesting?

Is that correct? The output of the factory is X kW, I believe.

 

[Vanadium 50]

Is that correct? The output of the factory is X kW, I believe.

I have the same question, but a different answer. The factory's output rate is measured in watts. The output over a period of time is measured in joules.

I'm trying to figure out what an E vs. t plot means so I can understand what the area under it might mean.

 

[Geoffrey Tullener]

Lets switch to joules and say that a battery hold 1J. A battery factory that churns out 20 batteries per hour is a 20 Jh facility.

I think the area represents the total about of energy capacity generated by the factory since production began. That would be a useful figure when trying to work out when it will pay back the carbon emissions used to create it. I think.

 

[DrClaude]

Lets switch to joules and say that a battery hold 1J. A battery factory that churns out 20 batteries per hour is a 20 Jh facility.

That makes it 20 J/h, not 20 Jh.

 

[anorlunda]

That makes it 20 J/h, not 20 Jh.

You're right. I made the same error.

 

[Geoffrey Tullener]

J/h is CJ/s is CWatts. That implies that battery factory converts materials and energy into power. That doesn't feel right. I think it converts those things into capacity which is a whole different set of equations. Maybe my example is not appropriate.

I think I confused things when I equated energy to energy capacity because that broke maths. I suspect our counterparts at Maths Forum are unhappy

 

[jbriggs444]

J/h is CJ/s is CWatts. That implies that battery factory converts materials and energy into power. That doesn't feel right. I think it converts those things into capacity which is a whole different set of equations. Maybe my example is not appropriate.

I think I confused things when I equated energy to energy capacity because that broke maths. I suspect our counterparts at Maths Forum are unhappy

The factory converts materials and energy into packaged energy. It does this at a certain rate over time. That's power.

 

[maxwells_demon]

Can anyone think of a use-case where ∫Edt is useful? ... and if such a thing has a name. Given that jerk snuck up on my in my 40s. Who would have though that the derivative of acceleration had a name? Does mom know?

My guess is it would be useful in climate change mitigation efforts.

Let's say coal mining. We often see how much coal is being mined in a year, which may be expressed in terms of mass. But with that mass is an energy equivalent, depending on what grade coal it is.

Let's say you were a researcher, and you wanted to find out how much (in terms of mass) that humanity has used up in terms of coal. And you only had data on the energy generated by coal-fired power plants in the entire planet, and which types of coal is burned by how many amounts. You then have meaningful use of your ∫Edt whose answer is directly convertible to mass of coal, and how much carbon we've been dumping to the atmosphere, as a spinoff. Carbon which would have otherwise been locked up had we not been using coal.

That same amount of carbon will be what we'll have to lock up in order to "reset" Earth to what would have been the present, had we found a cleaner alternative for all that energy consumption in the first place.

 

[jbriggs444]

∫Edt whose answer is directly convertible to mass of coal

That's the integral of energy per unit time d time. Not the integral of energy d time. It gives you a result in energy units which, as you point out, can be converted to mass units if we know the mass to energy conversion ratio.

 

[maxwells_demon]

That's the integral of energy per unit time d time. Not the integral of energy d time. It gives you a result in energy units which, as you point out, can be converted to mass units if we know the mass to energy conversion ratio.

oh yeah. mb. this is becoming some 4th dimensional dang.

 

[jbriggs444]

Let me try an exceedingly stupid way of making the integral of energy over time meaningful.

We have a very efficient power storage cell. It can store millions of joules, but is feather-light. We place it in deep space. Power cables link it back to earth. Nearby a pebble floats, stationary. Counter-weights are carefully arranged so that the pebble experiences no net gravitational attraction toward the storage cell.

Over time we pump the cell full of energy. And empty it. Mass-energy equivalence and gravity rear their heads. We want to know the resulting velocity of the pebble.

That velocity will relate to the integral of energy over time.

 

[maxwells_demon]

Can anyone think of a use-case where ∫Edt is useful? ... and if such a thing has a name. Given that jerk snuck up on my in my 40s. Who would have though that the derivative of acceleration had a name? Does mom know?

Ok this is actually trippy. Basically you put energy on a carrier. It just so happens that we got a perfect little carrier that we're familiar with. The charge. Energy per unit charge is voltage. The funny thing is the weber (unit of magnetic flux) is actually 1 volt-second. which means 1 Weber = 1 Joule-second per Coulomb

Therefore ∫Edt can be expressed in terms of weber-coulombs, and i don't know what that means. Our little ∫Edt may be found somewhere in the relationship between electricity and magnetism.

Therefore I conclude I'm too sleepy, and will hear yall's quirky and interesting ideas tomorrow.