electric fields - When work done is taken negative in electrostatics?

Let us say a point charge Q was moved across a potential difference V, then work done would be : QV.

This work is taken negative when done external agent, Please explain when It is negative and positive, also for the general case, if we take change in potential to be V, then by the equation :

When this will give an absolute value.

Answer

If a charge $Q$ is moved from rest by external forces across a potential difference $\Delta V=V_b-V_a$ from point $a$ to point $b$ and ends up at rest, then the net work done by all forces (electrical and external) will be zero by the work-energy theorem: $$W_{ext}+ Q(V_b-V_a) = \Delta K = 0.$$

That means that $W_{ext}=-Q(V_b-V_a)$. So, the work done by the field has the opposite sign of the external work.

If a positive charge is moved in the same direction as the electric field, or from higher to lower potential, the work done by the field will be positive.