# Potential Energy is defined only in conservative fields?

I am currently preparing for my physics test and struck in this question

Q. Potential Energy is defined:

1}only in conservative fields

2}As negative of work done by conservative forces

3}As negative of work done by external forces, when change in Kinetic energy is zero

As I read my study material it read :The work done by conservative forces is equal to decrease in Potential energy of system giving (2) as true and when its true then (3) also must be true by Conservation of mechanical Energy (I had done many question on it so related that with it )

But correct answer was (1) only

Can you help me out

• (2) and (3) give change in PE.
– Rick
Dec 7 '17 at 10:18
• @Rick MEANS THEY ARE CORRECT? Dec 7 '17 at 10:23
• Means they are incorrect. "x" and "the change in x" are two different things.
– Chris
Dec 7 '17 at 10:28
• Sorry didn't got it clearly , but are you saying its true for change in potential energy not for potential Energy?@Chris Dec 7 '17 at 10:38
• If the change in KE is zero then the external force is opposite (and equal) to the force exerted by the field. So the work done by external force is positive. So even if you think about change in PE, 3 is not correct.
– nasu
Oct 22 '18 at 14:40

You can check if you're dealing with a conservative field when you integrate the work along a closed path, a loop, you get zero because the integral is depending only on the starting point and finish point, which are the same. Also, in a conservative field, the sum of the kinetic and potential energies will be constant, no matter what path an object may take. You can imagine total energy being circulated between two reservoirs, one being potential and the other kinetic energy while the over all quantity stays the same.

• Does this answer the question? yesterday

A conservative force, $$\vec{F}= - \nabla \vec{V}$$. This implies that, for the conservative filed, force could be derived from the potential. For a non conservative field, the force could not be derived from the potential, as such potential exist in differential form and it would be impossible to write the potential explicitly as a function and hence the potential energy can only be defined for the conservative fields.

You might know that a direct consequence of Newton's second law ($$F=ma$$) is that the total change in kinetic energy of a body $$\Delta K$$ is equal to the total work $$W$$ of external forces, i.e.

$$\Delta K = W$$

If the difference in kinetic energy is zero ($$\Delta K = 0$$) then that implies

$$\Delta K = W = 0$$

so you get that the total work $$W$$ is 0. Clearly that implies (3) is false as it would mean the potential energy is 0.

We can split the work into the work coming from conservative forces $$W_c$$ and the work done by the rest of the forces $$W_R$$ so that $$\Delta K = W_c+W_R$$

Which means also that if $$W=0$$ the work of conservative force is equal and opposite to that of non-conservative forces ($$W_c=-W_R$$), and that's all we learn in this case.

From the above equation, we also get, shuffling the terms

$$-W_c= W_R-\Delta K$$

which clearly indicates that the negative work of the conservative forces is not the potential energy, so (2) is false.

The righ answer is (1), by exclusion.

Potential energy is indeed defined as the positive work done by conservative forces which in our case is $$W_c$$, so if we set $$W_c=U$$ then $$U=\Delta K - W_R$$ i.e. you could define potential energy also as the change in energy of your system minus the work done by non - conservative forces.

That is actually not the definition of potential energy but rather a consequence of it. The actual definition is $$U=W_c$$ or more in general, given a conservative force $$F$$ then if we go from a point in space $$A$$ to a point $$B$$ along a path $$\gamma$$ then the potential energy is the work down along that path by $$F$$.

The reason you need a conservative field to define a potential energy is that the above definition does not depend on the path your system takes from a point $$A$$ to a point $$B$$ so that whatever path you take $$U$$ is always the same. That is not true for $$W_R$$ or $$\Delta K$$ in general, it's only true for their difference (i.e. $$W_c$$).

From the above relationship and using $$W_c=U$$ i.e. $$\Delta K = W = U+W_R$$ we learn alternative definitions of $$U$$

1 - as the positive work done by conservative forces [not negative as in (2)]

2 - as the negative work done by non-conservative forces when $$\Delta K =0$$ because then $$U=-W_R$$ [not of the total work, as in (3)]

3 - as $$U = \Delta K -W_R$$

Hope this clear things up.