# A simple physics question …

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The following question was posed in New Scientist (13 Aug 2022) for readers to answer:  Does my electric vehicle weigh more after I have charged it?

Not surprisingly a couple of respondents wrote that because of Einstein’s equation  E = mc² the mass of the car will increase by a very small amount i.e. about 4 micrograms. Fair enough. No problem there.

But it got me thinking. Where is this extra mass?

Having done physics at degree level 40 years ago I feel I ought to be able to answer this question.  I’m sure you physics gurus will provide me an answer though.

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The extra mass is in the battery' or more specifically in the chemical potential energy stored in its charged state.

Regards Andrew

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1 minute ago, andrew s said:

The extra mass is in the battery' or more specifically in the chemical potential energy stored in its charged state.

Regards Andrew

ahh so that's why my phone keeps pulling my trousers down when I've charged it and pop it into my pocket, always wondered about that

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Posted (edited)
15 minutes ago, andrew s said:

The extra mass is in the battery' or more specifically in the chemical potential energy stored in its charged state.

Regards Andrew

I think therein lies my conceptual problem. I could see that the total mass would be higher if electrons had gained kinetic energy - because of their increased velocity. I think I’m finding it more difficult to see where the increased mass resides when there is an increase in potential energy.

Edited by Ouroboros
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If you have a 60 kWh battery in the EV, then when it is fully charged the energy in the battery will be 216 MJ (since 1 kWh is 3.6 MJ). So the change in mass of a fully charged battery compared to a discharged battery is only about 2.4 micrograms.

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As I see it, the battery has gained chemical energy.  The mass does not change.  If it did, you'd have both the energy increase due to mass increase, AND the chemical energy gain, which makes no sense to me.

Then there's the relativistic increase in mass due to speed.  Confusing yet again.  The amount of "stuff" does not increase, just the inertia.

I think......

Doug.

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21 minutes ago, Ouroboros said:

Does my electric vehicle weigh more after I have charged it?

It weighs more after charging it.

It has the same mass after charging it.

Weight of object is defined by acceleration in earths gravitation field. Acceleration depends on space time curvature. Energy and mass curve the space.

You increase energy content - but not mass. That is why object weights more - not because it gained mass. Rest mass remained the same.

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Posted (edited)
30 minutes ago, Ouroboros said:

I think therein lies my conceptual problem. I could see that the total mass would be higher if electrons had gained kinetic energy - because of their increased velocity. I think I’m finding it more difficult to see where the increased mass resides when there is an increase in potential energy.

Ok take a nucleus of an atom of Uranium 235 it is bound by the strong nuclear force. It has potential energy as if you it in a nuclear reactor the fission products have less rest mass energy than the original atom. The excess goes into the kinetic energy of the fission products which is captured to eventually make electricity.

It's the same with energy stored in chemical bonds or in separated charges as in a capacitor.

Regards Andrew

Edited by andrew s
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Posted (edited)
44 minutes ago, cloudsweeper said:

As I see it, the battery has gained chemical energy.  The mass does not change.  If it did, you'd have both the energy increase due to mass increase, AND the chemical energy gain, which makes no sense to me.

Doug.

They are two ways of expressing the same thing the amount of stored energy. See the example above on a Uranium nucleus.

In general relativity it is the total "stress energy tensor" that creates spacetime curvature.

Regards Andrew

Edited by andrew s
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Posted (edited)
44 minutes ago, vlaiv said:

It weighs more after charging it.

It has the same mass after charging it.

Weight of object is defined by acceleration in earths gravitation field. Acceleration depends on space time curvature. Energy and mass curve the space.

You increase energy content - but not mass. That is why object weights more - not because it gained mass. Rest mass remained the same.

I think this is a little confusing.  The rest mass of a Uranium 235 atom is greater than the sum of the rest masses of its fission products.  Binding energy of nuclei, atoms or molecules increases their rest masses.

Regards Andrew

PS see correction post below.

Edited by andrew s
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Posted (edited)
29 minutes ago, andrew s said:

I think this is a little confusing.  The rest mass of a Uranium 235 atom is greater than the sum of the rest masses of its fission products.  Binding energy of nuclei, atoms or molecules increases their rest masses.

Regards Andrew

On reflection that is not correct it depends on if the bound entity has  higher or lower energy than the components.  If energy is given out when they form it will have less energy and lower rest rest mass and if it takes energy to form them then it will have more energy and higher rest mass than the components.

In atomic nuclei lead is at the cross over point.

Sorry for the mistake.

Regards Andrew

Edited by andrew s
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7 minutes ago, andrew s said:

On reflection that is not correct it depends on if the bound entity has  higher or lower energy than the components.  If energy is given out when they form it will have less energy and lower rest rest mass and if it takes energy to form them then it will have more energy and higher rest mass than the components.

In atomic nuclei lead is at the cross over point.

Sorry for the mistake.

Regards Andrew

We could say that energy of atomic and molecular bonds is stored in gluon and photon quantum fields?

Gluons and photons don't really have rest mass, so all of that extra mass is due to relativistic effects of them whizzing about in nucleus and between electrons and nucleus.

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5 minutes ago, vlaiv said:

We could say that energy of atomic and molecular bonds is stored in gluon and photon quantum fields?

Gluons and photons don't really have rest mass, so all of that extra mass is due to relativistic effects of them whizzing about in nucleus and between electrons and nucleus.

To be honest I don't know if this is a good model or not. It seems plausible for something like Uranium 235 but for Helium it has a lower rest mass than its components but still has qluons holding it together.

Regards Andrew

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OK, I'm out of my depth, I admit!

What about stretching a spring?  The potential energy increases, but how can the mass increase?  Are there more iron atoms??

Doug.

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22 minutes ago, cloudsweeper said:

OK, I'm out of my depth, I admit!

What about stretching a spring?  The potential energy increases, but how can the mass increase?  Are there more iron atoms??

Doug.

No the number of iron atoms remains the same. What does change is the energy stored in the " bonds" holding the spring together.

Remember mass and energy are two faces of the same thing.

Unfortunately,  nature does not always fit our mental models of the world.

Regards Andrew

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6 minutes ago, andrew s said:

No the number of iron atoms remains the same. What does change is the energy stored in the " bonds" holding the spring together.

Remember mass and energy are two faces of the same thing.

Unfortunately,  nature does not always fit our mental models of the world.

Regards Andrew

Thanks Andrew.  No, I didn't really think there would be more atoms!

So the bonds change, but there is no more "stuff", so mass doesn't change.  Like the battery - the chemical bonds change, but the mass doesn't.

I agree about mental models.  We try to think of say light as a wave or a stream of particles, but in reality it is neither - or both.

Doug.

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Posted (edited)
24 minutes ago, cloudsweeper said:

So the bonds change, but there is no more "stuff", so mass doesn't change.  Like the battery - the chemical bonds change, but the mass doesn't.

Doug.

That's where you and nature disagree.

In the standard model all "stuff" is just excitations in the various quantum fields.

The more energy the more stuff.

Regards Andrew

Edited by andrew s
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1 hour ago, andrew s said:

To be honest I don't know if this is a good model or not. It seems plausible for something like Uranium 235 but for Helium it has a lower rest mass than its components but still has qluons holding it together.

Regards Andrew

Apparently quarks make just 1% of mass of nucleus - rest is down to gluon-quark dynamic (quarks moving, gluons moving and some other strange things).

It is probably down to geometry - takes more gluons to hold quarks in protons and neutrons when they are free particles than it does when they are "lumped together".

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3 hours ago, andrew s said:

Ok take a nucleus of an atom of Uranium 235 it is bound by the strong nuclear force. It has potential energy as if you it in a nuclear reactor the fission products have less rest mass energy than the original atom. The excess goes into the kinetic energy of the fission products which is captured to eventually make electricity.

It's the same with energy stored in chemical bonds or in separated charges as in a capacitor.

Regards Andrew

55 minutes ago, andrew s said:

No the number of iron atoms remains the same. What does change is the energy stored in the " bonds" holding the spring together.

Remember mass and energy are two faces of the same thing.

Unfortunately,  nature does not always fit our mental models of the world.

Regards Andrew

Yes, I think I (sort of) see - at least in terms of the equivalence of mass and energy in whatever form (kinetic or potential) it takes.

I think my ‘mental model’ is pretty naive, not to mention rusty. I’m afraid ‘quantum fields’  and ‘stress energy tensors’ go over my head somewhat.

Thanks for explaining anyway.

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To return to the original question.  It's easy to work out the amount of energy added to the battery.  But the mass of the battery does not increase.  Rather, by using E=mc^2, we can work out the mass equivalence of that increase in chemical energy.

Surely by the Conservation of Mass/Energy, there cannot be both an increase in chemical energy and an increase in mass?  Are the two not just mutually exclusive ways of describing the situation?

Doug.

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Posted (edited)
25 minutes ago, cloudsweeper said:

To return to the original question.  It's easy to work out the amount of energy added to the battery.  But the mass of the battery does not increase.  Rather, by using E=mc^2, we can work out the mass equivalence of that increase in chemical energy.

Surely by the Conservation of Mass/Energy, there cannot be both an increase in chemical energy and an increase in mass?  Are the two not just mutually exclusive ways of describing the situation?

Doug.

No they are mutually equivalent ways of saying the same thing. The increase in chemical energy is the the mass increase.

It's normally too small to bother about and only becomes significant with the strong nuclear force where it leads to nuclear power and weapons.

Regards Andrew

PS Try this

Edited by andrew s
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4 hours ago, andrew s said:

No they are mutually equivalent ways of saying the same thing. The increase in chemical energy is the the mass increase.

It's normally too small to bother about and only becomes significant with the strong nuclear force where it leads to nuclear power and weapons.

Regards Andrew

PS Try this

That’s a very nice little article isn’t it?  I don’t think I ever thought about all this in those terms, or at least not for a very very very long time.

That’s chemical reactions though. What about something more physical? Suppose I raise a 1kg mass through 100m in a uniform gravitational field. I will have done mgh Joules of work on that mass.  So will it have gained some mass in the process?  My feeling is that it must have gained mass because it now possesses more potential energy.  It’s more difficult to see where that mass is stored because, unlike a chemical reaction, there is no change in the configuration of chemical bonds within the object. Sorry if I’m making heavy weather of this.

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12 minutes ago, Ouroboros said:

That’s a very nice little article isn’t it?  I don’t think I ever thought about all this in those terms, or at least not for a very very very long time.

That’s chemical reactions though. What about something more physical? Suppose I raise a 1kg mass through 100m in a uniform gravitational field. I will have done mgh Joules of work on that mass.  So will it have gained some mass in the process?  My feeling is that it must have gained mass because it now possesses more potential energy.  It’s more difficult to see where that mass is stored because, unlike a chemical reaction, there is no change in the configuration of chemical bonds within the object. Sorry if I’m making heavy weather of this.

Simple answer is I don't know. Clearly,  moving the object changes the gravitational field so it might be stored in the gravitational field changes. I will do some research and get back to you unless someone answers before hand.

Regards Andrew

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Posted (edited)

Ok so here is the results of a quick search. Gravitational potential energy is not just a property of the raised mass but one between it and what is creating the gravitational field.

So to simplify,  assume two masses M and m initially together and you do work from outside the system to separate them then the total mass of the system of masses M and m will have increased by the small amount equivalent to the work done.

However, if M is the earth and it includes you and you throw a brick m upwards into orbit it won't have changed ! All you will have done is rearrange the internal energy of the system.

Regards Andrew

Edited by andrew s
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That seems to make sense. Or at least it will until I start thinking about it.

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