Calculating Work Done in Moving Point Charge: Potential vs. Potential Energy

[latentcorpse]

In my electrostatics notes, when calculating the work done in moving a point charge from r1 to r2 we use

W=\int_{\mathbf{r_1}}^{\mathbf{r_2}} \mathbf{F} \cdot \mathbf{dr}
which ends up giving W=qV where V is the potential difference.

However in previous years we used \mathbf{F}=-\nabla W \Rightarrow W=-\int_{\mathbf{r_1}}^{\mathbf{r_2}} \mathbf{F} \cdot \mathbf{dr}

(this works if you consider W as potential energy (mgz) and then you get a force due to gravity of -mg in the z direction.

so which is correct and why?

 

[dx]

Work done by what? The work done by you in moving a particle from r1 to r2 in a force field F is always

- \int_{{r_1}}^{{r_2}} \mathbf{F} \cdot \mathbf{dr}