Potential energy variation = work of -(conservative forces)

[AntoineCompagnie]

Homework Statement

Why is potential energy variation between two points equals to the work of the opposite of conservative forces between these two points?

Homework Equations

If we call these forces $$\vec F_ext^C$$

\begin{equation}
\Delta E_p=E_p(B)-E_p(A)=-\sum W_{A\rightarrow B}(\vec F_{ext}^C)
\end{equation}

The Attempt at a Solution

I thought it was the opposite of the kinetic energy...

 

[xOrbitz]

When we're dealing with conservative forces we know that all the energy before has to be equal to the energy after, that's why they're called conservative forces. Because of that we have
K_before + U_before = K_after + U_after
K_after - K_before = U_before - U_after
ΔK = -ΔU
However, the Work-Energy theorem tells us that ΔK = W, so that W = -ΔU
(_{K stands to kinetic energy, U stands to potential energy and W stands to work})

 

[AntoineCompagnie]

$\Delta K =- \Delta U$ is just because there was more kinetic energy before than after, isn't it?
And I'm dealing with Potenial energy, not kinetic one, does it change something?

 

[xOrbitz]

$\Delta K =- \Delta U$ is just because there was more kinetic energy before than after, isn't it?
And I'm dealing with Potenial energy, not kinetic one, does it change something?

Think about you just said: K_after < K_before ∴ K_after - K_before < 0, if ΔK < 0 then ΔU > 0, right? So what does it means? It means that when the variation of kinetic energy decrease then the variation of potential energy will increase, because it must conserve energy, just think about the energy conservation when dealing with conservative forces.
So in fact you can say either ΔK = W or ΔU = - W since ΔK + ΔU = 0, in other words, W - W = 0 (which holds).
But it's really really important for you to remember that this is only true when there's energy conservation, hence the external force is conservative.