Change in Potential Energy and it's Contradictory Signs

Question

Ok, consider this sign convention for change in potential energy.

If $ \Delta U = + $ : There is increase in potential energy
If $ \Delta U = - $ : There is decrease in potential energy

Now, lets's move onto the main part of the post.

Consider 2 point positve charges, Q as a source charge and q as the test charge.
Now, the Q charge will have electric field, $\vec{E}$ going towards to $\infty$.
The datum is considered $\infty$ where $ U_{\infty} = 0 $.

The point charge, q is brought from $ \infty $ to a distance, r from Q charge. While bringing it from $ \infty $ to a distance, r, an external force, $ \vec{F} $ acts on it which is equal and opposite to the electric force $ \vec{F_e} $

So, $ \vec{F} = -\vec{F_e} $ and $ F_e = k \frac{Qq}{r^2} $

Let the work done by the electric force be $ \,dW_e $. So,

$$ \int dW_e = \int_{\infty}^r \vec{F_e}\,d{\vec{r}} $$ $$ W_e = q\int _{\infty}^r \vec{E}\, d{\vec{r}} $$ $$ W_e = -q\int_{\infty}^r E\,dr $$ $$ W_e = -kQq\int_{\infty}^r \frac{1}{r^2}\, dr $$ $$ W_e = -kQq\biggl[\frac{-1}{r}\biggr]_{\infty}^r $$ $$ W_e = k\frac{Qq}{r} $$

Now, as $ \vec{F_e} $ is a conservative force, so

$ W_e = -\Delta U $

Thus,

$$ {\color{red} {\Delta U = -k\frac{Qq}{r}}} $$

Now, Q, q, r : All these variables are positive.
So,

$ \Delta U = Negative$

Thus, this means that Final Potential Energy < Initial Potential Energy or to put in other ways potential is decreased when it is moved from $ \infty $ to distance ,r.

Now, this is supposed to be a contradictory result. Why??

Because one can easily see that $\vec{E} $ is going towards $ \infty $, the direction of decreasing potential/potential energy.
And because, $ U_\infty = 0 $ so, U_r should certainly be positive.

But this is not what the result is telling me. Why such contradiction is there in the result?

Using same sign conventions do not create any problems in case of negative source charges ,i.e, -Q.

So, please help in clearing this doubt. Also, please go through the next section.

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Some Extra Points Related To This

-----> First of all to notice is that the angle between $ \vec{E} $ and $ \, d{\vec{r}} $ is 180°.

The angle is 180° because there is an external force, $ \vec{F} $ acting equal and opposite to the electric force, $ \vec{F_e} $ along which the charge, q moved(i.e., displacement is opposite to electric force).

The purpose of external force is to just balance the electric force so that there is no net increase in kinetic energy of the test charge. Otherwise, in the absence of external force, while evaluating the work done, $ W_e $, I had to consider the net kinetic energy also which would also change the expected value of potential energy at the distance, r.

-----> The most expected answer I think, I will get is that "you have set the parameter/sign conventions of potential energy like that". If you swap the signs, everything will get resolved.

Ok, see, if I even swap the signs of potential energy given at the very beginning of the post, it will resolve the problem/contradiction ocurring.

But, if I swap the signs what it will do that in case of negative source charges(i.e., -Q) same contradiction will occur but the signs will get reversed in the value of potential energy.

In that case, the external force will move the positive test charge, q from distance, r to $ \infty $ opposite to the direction of attractive electric force $ \vec{F_e{_A}} $. So the limits would be like $ \int_r^{\infty} \vec{F_e{_A}}\, d{\vec{r}} $

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Lastly, I know there are many experts in this community. No disrespect, but please don't write that "the sign of change in potential energy don't matter at the end".

According to me the sign of change in potential energy matters as long as you are stuck with one set of sign conventions but the sign of potential energy(at a point) doesn't matters. If I am wrong at this point, please correct me.

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Edit

In case of +Q charge, the external force will move the test charge,q from $\infty\rightarrow r$ whereas for -Q charge, it will take it from $r\rightarrow \infty$.

Due to the conservative nature, the electric force does 0 work in closed path. Thus,

$W_{\infty\rightarrow r}=- W_{r\rightarrow \infty}$

Considering Bill N statement that the differential element $d{\vec{r}}$ is always points from origin to $\infty$ then,

For +Q charges, the repulsive force $\vec{F_e}$ is parallel to $d{\vec{r}}$ and for -Q charges, the attractive force $\vec{F_e}$ is anti-parallel to $d{\vec{r}}$.

Now, as the direction of differential element is handled by limits of integration(Bill N comment) then, it's for sure that for either of the 2 cases(+Q and -Q source charges), the scalar product between $\vec{F_e}$ and $d{\vec{r}}$ should be same.

Earlier, in this post, for +Q charges, the work done by repulsive force was coming out to be $W_{\infty\rightarrow r} = k\frac{Qq}{r}$ and so the $\Delta U= -k\frac{Qq}{r}$

A counter-intuitive result if $\Delta U= U_{final}-U_{initial}$ While evaluating the $W_{\infty\rightarrow r}$ the negative sign in the result of integration was cancelled out by the negative sign of scalar product.

I have done the same for -Q charges and the expression was $W_{r\rightarrow \infty}= -k\frac{Qq}{r}$ and so the $\Delta U= k\frac{Qq}{r}$.

Again, an invalid result if $\Delta U= U_{final}-U{initial}$. Here, the negative sign of scalar product was the only negative sign.

I did all of this was done again but taking angle between the $\vec{F_e}$ and $d{\vec{r}}$ equal to 0° for both types of source charges. This time, the results were opposite and signs of change in potential energy were absolutely correct(if $\Delta U= U_{final}-U_{initial}$).

So, the only thing which is striking my mind is why the angle between the 2 when taken 0° is giving right signs in $\Delta U$ even though for negative source charges, the $d{\vec{r}}$ is opposite to $\vec{F_e}$?

Answer 1

Your mistake is in the third integral equation and your consequent statement later:

-----> First of all to notice is that the angle between E⃗ and dr⃗ is 180°.

No, it's not! When you set up $\int \vec{E}\cdot \mathrm{d}\vec{r}$ you must keep in mind that the $\mathrm{d}\vec{r}$ must be consistent with the limits, too. The ordering of the limits, not the differential vector element, tells you what direction you are moving when you do the integral. Let's consider the following 2 integrals $$\int\limits_3^6 x ~\mathrm{d}x \hat{i} \text{ and} \int\limits_6^3 x~ \mathrm{d}x \hat{i}.$$

They must be the negative of each other. So, the ordering of the limits determines what the integral means. The differential element is necessarily interpreted as defining the positive change direction for the increment.

In other words, $\mathrm{d}\vec{r}$ points out from the origin toward $+\infty$ and is parallel to $\vec{E}$. That's where your incorrect negative sign comes from.

Answer 2

Ok, I think I got it but it may become hard for me to explain.
Things gets started to wrong when I assumed that Final Potential Energy < Initial Potential Energy.

It means that $\Delta U = U_f - U_i $.
But if I take $\Delta U = U_i - U_f $ then everything goes right with the situation, i.e., Final Potential Energy > Initial Potential Energy and thus, no contradictions.

See, the point is that the sign of $\Delta U $ you will get entirely depend on the limits of integration.
Swap the limits and the sign of $\Delta U $ will get reversed. For more information relating to the swapping of limits of integration refer to this page enter link description here

So, the sign in the change of potential energy, $\Delta U $ doesn't explicitly concludes that there is increase in the potential energy or decrease in potential energy(which I what I did in my question).

The conclusion that there will be increase or decrease of potential energy between any 2 points depends how you take the difference between the potential energy of 2 points(it could be $ U_f-U_i $ or $ U_i-U_f $).

Also, while finding the potential energy at a point, either one of them($U_f$ or $U_i$) may be the datum/reference point.