What is the equation for gravitational and electrical potential energy?

[rogerk8]

Hi!

I just wish you to help me derive two types of potential energy.

A) Gravitational Potential Energy
B) Electrical Potential Energy

A useful equation is

$$W=\int Fdr$$

Which I consider is the work that has to be done to the particle to move it a certain distance from the attractive centre of force (which then becomes it's PE, is my thought).

For A we have

$$F=G\frac{mM}{r^2}$$

Now

$$W=\int_a^hG\frac{mM}{r^2}=GmM\int_a^h \frac{1}{r^2}$$

Which means

$$W=GmM((-1/h)-(-1/a))=GmM(1/a-1/h)$$

Where a cannot be zero but maybe it could be R?

Then we have

$$W=\frac{GmM}{R}(1-\frac{R}{h})$$

And if we set

$$\frac{GM}{R}=g$$

We have

$$W=mg(1-\frac{R}{h})$$

But this till does not look like Ep=mgh :D

I get the same problem in the B-case.

Considering Hydrogen for simplicity

$$F=\frac{e^2}{kr^2}$$

then

$$W=\int_a^h \frac{e^2}{kr^2}=\frac{e^2}{k}\int_a^h \frac{1}{r^2}$$

which gives

$$W=\frac{e^2}{k}((-1/h-(-1/a))=\frac{e^2}{k}(1/a-1/h)$$

where once again a cannot be zero.

Can it be the shell radius, then?

There's no real use starting at a point inside the shell radius so let's play with r, this time.

This gives

$$W=\frac{e^2}{kr}(1-r/h)$$

Which is about the same result as for the Gravitational Potential Energy above.

Is this totally wrong?

Roger
PS
It is interesting to note that PE increses with distance in both cases. If I'm right, that is :)

 

[DEvens]

What is $h$? Is it supposed to be the height above the original location? The height above the surface of the Earth? The distance from the location a?

If I guess that $h$ is supposed to be the distance from $a$ to $b$, then $r$ at $a$ is $a$, right? And $r$ at $b$ is $b=a+h$, not $h$. Right?

So I think you got your integral a little wrong. I think you did something similar in the electric force case.

After that, the formula $E_p = mgh$ assumes a constant gravity force. To get that, you would need to use the fact that $h << R$ where $R$ is the radius of the surface of the Earth, and you are not moving very far from the surface.

If $h$ is large enough that you cannot consider the force to be constant over a change of location the size of $h$, then you should not get a potential proportional to $h$.

 

[Dale]

Where a cannot be zero

Usually we set $a=\infty$

 

[rogerk8]

What is $h$? Is it supposed to be the height above the original location? The height above the surface of the Earth? The distance from the location a?

I think it is quite obvious that h is something larger than a, otherwise the integral sign would be reversed.

If I guess that $h$ is supposed to be the distance from $a$ to $b$, then $r$ at $a$ is $a$, right? And $r$ at $b$ is $b=a+h$, not $h$. Right?

Here you are being a bit technocratic while of course being right, upper integration limit should equal a+h. I however just see it like h is "much" larger than a (in the EPE case) so you might as well use h only. This is however not true for GPE, I now understand.

So I think you got your integral a little wrong. I think you did something similar in the electric force case.

Sloppy of me :)

After that, the formula $E_p = mgh$ assumes a constant gravity force. To get that, you would need to use the fact that $h << R$ where $R$ is the radius of the surface of the Earth, and you are not moving very far from the surface.

If $h$ is large enough that you cannot consider the force to be constant over a change of location the size of $h$, then you should not get a potential proportional to $h$.

A kind of obvious statement but thank you for telling me.

Thank you for your reply!

I will now revise my integrals :)

Roger

 

[Dale]

Now

$$W=\int_a^hG\frac{mM}{r^2}=GmM\int_a^h \frac{1}{r^2}$$

You are probably actually interested in:

$$W=\int_R^{R+h}G\frac{mM}{r^2} dr$$

which gives (you can confirm this by substituting in these limits in the expression you derived above)
$$\frac{GMmh}{R^2+hR}$$

Do a Taylor series expansion around h=0 to get
$$\frac{GMmh}{R^2} + O(h^2) \approx mgh$$

for $g=GM/R^2$

 

[DEvens]

Dale, I was trying not to do the entire question for him.

 

[Dale]

Oh, I thought you were unsure too. Oops.

 

[rogerk8]

Hi!

I just wish you to help me derive two types of potential energy.

A) Gravitational Potential Energy
B) Electrical Potential Energy

A useful equation is

$$W=\int Fdr$$

Which I consider is the work that has to be done to the particle to move it a certain distance from the attractive centre of force (which then becomes it's PE, is my thought).

For A we have

$$F=G\frac{mM}{r^2}$$

Now

$$W=\int_a^{a+h}G\frac{mM}{r^2}dr=GmM\int_a^{a+h} \frac{1}{r^2}dr$$

Which means

$$W=GmM((-1/(a+h))-(-1/a))=GmM(1/a-1/(a+h))$$

Where a cannot be zero but maybe it could be R?

Then we have

$$W=GmM(\frac{1}{R}-\frac{1}{R+h})$$

By setting

$$\frac{GM}{R}=g$$

We have

$$W=mg(1-\frac{R}{R+h})$$

But this still does not look like Ep=mgh :D

For fun sake we may rewrite this as

$$W=\frac{m^2}{k'R}(1-\frac{R}{R+h})$$

I get the same problem in the B-case.

Considering Hydrogen for simplicity

$$F=\frac{e^2}{kr^2}$$

then

$$W=\int_a^{a+h} \frac{e^2}{kr^2}dr=\frac{e^2}{k}\int_a^{a+h} \frac{1}{r^2}dr$$

which gives

$$W=\frac{e^2}{k}((-1/(a+h)-(-1/a))=\frac{e^2}{k}(1/a-1/(a+h))$$

where once again a cannot be zero.

Can it be the shell radius, then?

There's no real use starting at a point inside the shell radius so let's play with r, this time.

This gives

$$W=\frac{e^2}{kr}(1-\frac{r}{r+h})$$

Which is about the same result as for the Gravitational Potential Energy above.

Is this totally wrong?

Roger
PS
It is interesting to note that PE increses with distance in both cases and that it at infinity approaches "shell" (r or R radius) potential energy

Can this be in the vicinity of being right?

Roger

 

[rogerk8]

Let's try:

$$W=GmM(\frac{1}{R}-\frac{1}{R+h})$$

$$W=GmM\frac{(R+h)-R}{R(R+h)}$$

$$W=GmM\frac{h}{R(R+h)}$$

I got it :D

I think I get your Taylor expansion too, DaleSpam!

While h is so small copared to R, h^2 terms does not matter.

But it is really a more approximate formula than I thought!

Roger

 

[Dale]

But it is really a more approximate formula than I thought!

Yes. The $mgh$ formula is an approximation. Of course, the $GMm/r^2$ formula is also an approximation. The corresponding general relativity equations are also approximations.

 

[rogerk8]

I've decided to collect my formulas so they look nice and comprehensive:

This is an attempt in deriving two expressions for potential energy.

A) Gravitational Potential Energy (GPE)
B) Electrical Potential Energy (EPE)

A useful equation is

$$W=\int Fdr$$

Which I consider is the work that has to be done to the particle to move it a certain distance from the attractive centre of force (which then becomes it's PE, is my thought).

For A we have

$$F=G\frac{mM}{r^2}$$

Now

$$W=\int_a^{a+h}G\frac{mM}{r^2}dr=GmM\int_a^{a+h} \frac{1}{r^2}dr$$

Which means

$$W=GmM((-1/(a+h))-(-1/a))=GmM(1/a-1/(a+h))$$

Where a cannot be zero but maybe it could be R?

Then we have

$$W=GmM(\frac{1}{R}-\frac{1}{R+h})$$

Rearrangeing this equation we get

$$W=GmM\frac{(R+h)-R}{R(R+h)}$$

and finally

$$W=GmM\frac{h}{R(R+h)}$$

which for h<<R gives

$$W=\frac{GmMh}{R^2}$$

and putting

$$g=\frac{GM}{R^2}$$

gives

$$W=mgh$$In the B-case we have:

Considering Hydrogen for simplicity

$$F=\frac{e^2}{kr^2}$$

then

$$W=\int_a^{a+h} \frac{e^2}{kr^2}dr=\frac{e^2}{k}\int_a^{a+h} \frac{1}{r^2}dr$$

which gives

$$W=\frac{e^2}{k}((-1/(a+h)-(-1/a))=\frac{e^2}{k}(1/a-1/(a+h))$$

where once again a cannot be zero.

Can it be the shell radius, then?

There's no real use starting at a point inside the shell radius so let's play with r, this time.

This gives In the same manner as for GPE above

$$W=\frac{e^2}{k}\frac{h}{r(r+h)}$$

or when h>>r

$$\frac{e^2}{kr}$$

Let's repeat two equations for convenience:

$$W=GmM\frac{h}{R(R+h)}$$

$$W=\frac{e^2}{k}\frac{h}{r(r+h)}$$

The first is GPE non-approximated, the second is EPE non-approximated.

It is interesting to note that they only differ in two aspects:

1) Different constants.
2) In the GPE case the normal use is for h<<R while in the EPE case the normal use is for h>>r

It is even more interesting to note that PE increases with h.

Roger

 

[rogerk8]

It is a bit fascinating what happens in extreme cases of GPE and EPE.

If we begin with GPE.

When h is much larger than R we have

$$Ep=\frac{mMG}{R}$$

Which means that the potential energy of a mass unit outside the gravity field cannot be higher than that at the surface of (earth).

And in the EPE case where h (or r) is much larger than the nuclei radius we have

$$Ep=\frac{e^2}{kr}$$

So the potential energy is related to the electron orbit radius when Ep is considered inside the atom, but the same if the electron is far from the electron radius.

This has to do with the same principle as above that is if the electron is far from its orbit, its energy is what I say but we could turn that around and say that if the electron really is in its orbit, the distance to the nuclei radius is "equally far" so the same principle may apply.

Right?

Roger

 

[Dale]

Which means that the potential energy of a mass unit outside the gravity field cannot be higher than that at the surface of (earth).

No, in that formula the PE = 0 at h = 0. Indeed, that is the meaning of setting the lower limit of the original integral to R. You are choosing to define the PE such that it is 0 at R.

 

[rogerk8]

No, in that formula the PE = 0 at h = 0. Indeed, that is the meaning of setting the lower limit of the original integral to R. You are choosing to define the PE such that it is 0 at R.

I don't agree.

In my humble opinion GPE in the farfield is exactly that as at the surface, R

Repeating for convenience

Nearfield GPE (NGPE):

$$Ep=GmM\frac{h}{R(R+h)}$$

Farfield GPE (FGPE):

$$Ep=\frac{mMG}{R}$$

If you look at NGPE ant use h>>R you get FGPE.

In FGPE the only variable (except m) is R and R is the radius of the Earth.

So I would call FGPE as I said, that is that a mass unit far from the Earth cannot have higher PE than what it has on the surface, thus R.

Sorry, I see now that I am wrong because at small h I need to use NGPE which is zero at h=0 (as you say).

It was just the fact that only R is present in the FGPE expression that fooled me.

Also, I have made terrible mistakes regarding EPE too.

I do not even want to go into them.

Well, I do :)

Repeating for convenience

Nearfield EPE (NEPE):

$$W=\frac{e^2}{k}\frac{h}{r(r+h)}$$

Farfield EPE (FEPE):

$$Ep=\frac{e^2}{kr}$$

Here we have chosen the boundaries to be from the electron orbital radius (r) to a hight (r+h).

In the NEPE-case, h is small (compared to r).

In the FEPE-case, h is large.

Both equations considers movement of electron outside of its shell.

But what happens if we try to use the equation inside the shell?

Well, in that case we have

$$r=r_n$$

and

$$h=r_e$$

Where n stands for nuclei and e for electron.

If we just imagine this we may reuse the formula for FEPE (while h>>r)

So that the PE of the orbiting electron becomes

$$Ep=\frac{e^2}{kr_n}$$

Which probably is not true :)

Roger
PS
Wait a minute. What is the potential energy of an electron doing what it's born to do? Maybe we could set EPE to zero for h=0 in the same manner as we do for GPE?

 

[Vanadium 50]

I don't agree.

In my humble opinion GPE in the farfield is exactly that as at the surface, R

First, physics doesn't care about your opinion. Second, you're wrong.

 

[rogerk8]

First, physics doesn't care about your opinion. Second, you're wrong.

Have you read my whole post?

Where I confess that I understand that I'm wrong?

Or is this the way you have made over 18000 posts?

 

[jtbell]

I don't agree.

In my humble opinion GPE in the farfield is exactly that as at the surface, R

No, this is wrong.

Repeating for convenience

Nearfield GPE (NGPE):

$$Ep=GmM\frac{h}{R(R+h)}$$

Farfield GPE (FGPE):

$$Ep=\frac{mMG}{R}$$

Both of these are OK. Now, at the surface of the earth, h = 0, so: $$E_p = GmM \frac{0}{R(R+0)} = GmM \frac{0}{R^2} = 0$$ which is not the same as the far-field GPE ($h \rightarrow \infty$).

 

[rogerk8]

No, this is wrong.
Both of these are OK. Now, at the surface of the earth, h = 0, so: $$E_p = GmM \frac{0}{R(R+0)} = GmM \frac{0}{R^2} = 0$$ which is not the same as the far-field GPE ($h \rightarrow \infty$).

Thanks for correcting me!

Roger

 

[Vanadium 50]

Have you read my whole post?

Where I confess that I understand that I'm wrong?

No, I didn't. Once people say something wrong, there is no point in reading further. Because most people don't hit the "post reply" but after they have written what they themselves recognize is uttrer and complete nonsense, because most people are respectful enough of others to not deliberately waste their time. I was wrong, obviously.

 

[rogerk8]

You are forgiven Vanadium 50.

Thank you for getting back to me!

But I hope it is ok to elaborate own thoughts which may not be considered physics truths but just to lay them on the table.

I actually think this is a liberating way to attack a problem.

You state something, you reflect upon what you have said, you either stick to it or abandon it.

I think this way of reasoning is rather healthy.

But of course, I can always listen to you skilled guys telling me how things are.

But try and put yourself into my ignorant position, how much will I then really understand?

Roger

 

[Dale]

Nearfield GPE (NGPE):

$$Ep=GmM\frac{h}{R(R+h)}$$

Farfield GPE (FGPE):

$$Ep=\frac{mMG}{R}$$

If you look at NGPE ant use h>>R you get FGPE.

In FGPE the only variable (except m) is R and R is the radius of the Earth.

So, I think that this may be an important misconception. R is not necessarily the radius of the earth. R is an arbitrary reference radius. At that reference radius you have defined h = 0 and also GPE = 0. Often R is chosen as the surface of the earth, but there is no reason that it could not be some other radius, or even infinity.

There is also no reason that h = 0 needs to be defined at the same location as GPE = 0. In fact, the other most common convention is to measure h from the center (usually denoted "r" instead of "h") and set GPE = 0 at infinity.

Also, the equation that you have labeled NGPE is just GPE. It covers both "near" and "far" using the above-mentioned convention for h and R.

I actually think this is a liberating way to attack a problem.

It may be liberating for you, but it is also a time-sink for those responding who will have difficulty in determining what would be most helpful to you. As it is, I have no idea if my comments above are at all relevant or if you already figured it out.

 

[nasu]

But I hope it is ok to elaborate own thoughts which may not be considered physics truths but just to lay them on the table.

I actually think this is a liberating way to attack a problem.

You state something, you reflect upon what you have said, you either stick to it or abandon it.

It definitely is a good way to do this for yourself.

But once you realize that something has to be abandoned, that is was the wrong path, it what is the point to post it here? You already learned (supposedly) from it.
You may record this in your notes or journal, but to "publish" all your (known) errors looks a little self centered. Especially when you do it in long posts.

 

[rogerk8]

So, I think that this may be an important misconception. R is not necessarily the radius of the earth. R is an arbitrary reference radius. At that reference radius you have defined h = 0 and also GPE = 0. Often R is chosen as the surface of the earth, but there is no reason that it could not be some other radius, or even infinity.

Ok

There is also no reason that h = 0 needs to be defined at the same location as GPE = 0. In fact, the other most common convention is to measure h from the center (usually denoted "r" instead of "h") and set GPE = 0 at infinity.

I verbally understand what you are saying, but I do not fully grasp it.

What would this integral look like?

Doing work along the gravitational force, I mean.

What would that give?

Accelerated work?

I am serious with this while not understanding so much :)

Also, the equation that you have labeled NGPE is just GPE. It covers both "near" and "far" using the above-mentioned convention for h and R.

Of course it is and does, I just think it is easier to actually plain write what GPE is at far-field also (beacause it may be obvious to you but it certainly is not obvious to me and other ignorants).

It may be liberating for you, but it is also a time-sink for those responding who will have difficulty in determining what would be most helpful to you. As it is, I have no idea if my comments above are at all relevant or if you already figured it out.

Your comments where perfect and I thank you for them.

Otherwise, put yourself into my position. I am ignorant when it comes to physics. I have some basic understanding though. But most of all, I have a passionate interest (in parts of physics, anyway).

So while I'm ignorant, what do you suggest I do?

Just write questions and wait for you to answer them?

Or something in between, that is trying to solve the problem by yourself while chatting with you nice guys?

Which is more fun?

I am an extremely lonely guy, so what do you think I prefere?

One more thing, putting something on the table simplifies all kind of discussions.

I even think that many people do not dare to put basic questions on the table.

They are afraid that they would look stupid.

Actually I know that Ericsson here in Sweden actually has special employees who are employed for the sole purpose to ask "stupid" questions at meetings.

People are too proud (and afraid).

Finally, to answer nasu above at the same time, I am not the least self centered. I just lack self confidence.

And I'm trying desperatelly to build it up here at PF, if you don't mind!

The start of my intrinsic self confidence will be that I will write things in my way nomatter what you guys think.

With respect, however.

Take care!

Roger

 

[nasu]

Ok
Finally, to answer nasu above at the same time, I am not the least self centered. I just lack self confidence.

And I'm trying desperatelly to build it up here at PF, if you don't mind!

The start of my intrinsic self confidence will be that I will write things in my way nomatter what you guys think.

With respect, however.

Take care!

Roger

Maybe lack of self confidence has different meaning in Sweden.
But I respect that, too.

 

[Dale]

I verbally understand what you are saying, but I do not fully grasp it.

What would this integral look like?

Like this:
$$W=\int_{\infty}^r \frac{GMm}{r^2} dr$$

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html#ui

Note that $W=0$ for $r=\infty$, so the reference location is not $r=0$. Also, note that $W<0$ for all $r$, so the GPE is negative. Any particle whose KE is more positive than its GPE is negative will still have energy left over when it "gets to infinity". Such an object is said to be able to escape and the corresponding speed is called escape velocity.

 

[rogerk8]

Like this:
$$W=\int_{\infty}^r \frac{GMm}{r^2} dr$$

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html#ui

Can you really do that?

I mean set the lower limit to infinity and just integrate to arbitrary r?

You have the gravitational force equation in the integral, right?

This formula tells the force at different distances, right?

To use this formula and the work that is done to m (which gives its Ep) would mean movement against the gravitational force.

Now your telling me that you might as well take m at infinity and move it along the gravitational force (towards r).

But this is not work, thus not Ep in my book.

To try to be a little confident ;)

Let's solve the integral:

$$W'=GMm((-1/r)-(-1/\infty))=-\frac{GmM}{r}$$

Let's say that this really is the potential energy, Ep.

What is then r?

I think it is as you say, the distance to the core (or hight over the core).

Because we could easily imagine M to be tiny so that r needn't be so large while being apart from M.

M need not have the radius of R, right?

It may be, I have no clue about these things other than that they are dense, a neutron star.

So if M is a neutron star, r may actually approach zero.

In other words, we have an expression that covers the whole space, actually.

Ep is zero at infinity, but has a value

$$Ep=\frac{mMG}{r}$$

close to M.

I do however think it is more convenient to have Ep=0 at the surface of the Earth and

$$Ep=\frac{mMG}{R}$$

at infinity (or>>R)

Roger

 

[Qwertywerty]

Can you really do that?

I mean set the lower limit to infinity and just integrate to arbitrary r?

Yes , you can .

You have the gravitational force equation in the integral, right?

Yes .

This formula tells the force at different distances, right?

No . It tells you how much work , at some distance x ( ∀ all x ∈ ( -∞ , r ) ) , it does on moving a small distance dx .
Remember , dW = F . ds ( formula for differential work ) .

 

[Qwertywerty]

What is then r?

It refers to the distance at which you wish to find the potential energy .

Because we could easily imagine M to be tiny so that r needn't be so large while being apart from M.

M need not have the radius of R, right?

It may be, I have no clue about these things other than that they are dense, a neutron star.

So if M is a neutron star, r may actually approach zero.

I didn't gat this .

Ep is zero at infinity, but has a value

Ep=mMGr​

Ep=\frac{mMG}{r}

close to M.

The -ve sign is important , you know .

I do however think it is more convenient to have Ep=0 at the surface of the Earth and

Ep=mMGR​

Ep=\frac{mMG}{R}

at infinity (or>>R)

Potential energy does not have any fixed value anywhere , and is always defined with respect to another point in space .
That being said , potential energy as zero at infinity is a very conventional norm .

 

[Dale]

Can you really do that?

I mean set the lower limit to infinity and just integrate to arbitrary r?

Yes.

M need not have the radius of R, right?

Correct, R does not even enter into the equation.

I do however think it is more convenient to have Ep=0 at the surface of the Earth and

Ep=mMGR​

Ep=\frac{mMG}{R}

at infinity (or>>R)

Sometimes one will be more convenient, and sometimes the other will. You should know both so you can always pick the one that is most convenient for a given situation

 

[rogerk8]

I think I get this now!

Thank you all for helping me!

Ep is arbitrary chosen, right?

It is not fixed like Ek, right?

It is based on a reference point.

This is total news to me!

News and strange because the way I look at Ep is "the work to get there".

But if Ep=0 at infinity, Ep might as well be negative, right?

Because we could say that "we have already moved it to infinity, now we wish to move it back (and thus against the force)".

This is very interesting (hope I'm right).

Summarizing my formulas:

GPE:

$$Ep=GmM\frac{h}{R(R+h)}$$

Nearfield or Normal GPE (NGPE):

$$Ep=mgh$$

Farfield GPE (FGPE):

$$Ep=\frac{GmM}{R}$$

EPE:

$$Ep=-\frac{k'e^2}{r}$$

where

$$k'=\frac{1}{4\pi \epsilon_0}$$

The interesting thing here is that if we compare FGPE with EPE the major difference is the sign.

Both has the exact same structure but differs in the sign.

I have chosen to use your tip DaleSpam and converted it to the most useful situation, EPE.

Because in that case it is more convenient to have Ep=0 at infinity and some value at smaller distances.

That way we are able to calculate Ep anywhere.

Still, "someone" has moved the charge to infinity...

GPE as stated above may always be used and I think it is the most appropriate formula due to Ep=0 at the Earth's surface.

NGPE and FGPE are approxiamtions of GPE.

In most cases, NGPE suffices, but for objects high up in the sky, FGPE is a good approximation.

Thanks for listening :)

Roger
PS
I made a couple of mistakes above:

1)

"This formula tells the force at different distances, right?"

Should read "This equation tells the force at different distances, right?"

2)

$$Ep=\frac{mMG}{r}$$

Should have a minus sign in it.

 

[Dale]

Ep is arbitrary chosen, right?

It is not fixed like Ek, right?

It is based on a reference point.

Umm, KE is also not fixed. It is based on a reference frame. I am sitting on my chair, so in the Earth's reference frame my KE is 0, but in the Sun's reference frame I am moving so I have a substantial KE.

Energy is conserved regardless of which reference frame you use for KE and what reference point you use for PE, but different reference frames will give different values for KE and different reference points will give different values for PE.

But if Ep=0 at infinity, Ep might as well be negative, right?

Yes, if you choose the common convention of setting gravitational PE to 0 at infinity then at all finite distances the PE is negative. There is nothing wrong with that.

The interesting thing here is that if we compare FGPE with EPE the major difference is the sign.

Both has the exact same structure but differs in the sign.

I haven't been following your electrical PE stuff. Just like there are two masses in the gravitational equation there should also be two charges in the electrical one. If they have the same sign then the overall expression (using PE = 0 at infinity) is positive, indicating that it is repulsive. If they have opposite signs then the expression is negative, indicating that it is attractive. So the sign is folded into the charge. When the force is attractive, like gravity, you automatically get a negative sign for the PE, like gravity.

 

[Radhakrishnam]

'And if we set

GM/R=g'

You can see this setting is incorrect; the units do not balance. Consequently, your result for W also doesn't give the right equation.
In the limits of integration, if you put (R+a) and (R+h) in place of a and h, you get:
W= mgR² [(R+a)^-1 - (R+h)^-1]
Neglecting R(a+h) and (ah) in comparison with R², we get:
W= mg(h - a).

I hope this would be helpful.
P. Radhakrishnamurty