Electric Potential
Question 1. Is $V=W/Q$ or $V=P.E./Q$.
P.E. is electric potential energy
Potential difference
Question 2. Is $\Delta$$V=W/Q$ or $\Delta$$V=$$\Delta$$P.E./Q$.
Question 3. Is there any relationship between work and potential energy in this case?
EDIT:
Why change in Electric Potential Energy is equal to the work done?
Consider a region of space with an static electric field $\mathbf{E}$, Now if I displace a charge (unit charge) from one point to other then the work done by the force given by $$V(\mathbf{r}_b)-V(\mathbf{r}_a)=-\int_{\mathbf{r}_a}^{\mathbf{r}_b}\mathbf{E}\cdot d\mathbf{r}=-W_{ba}$$
Where $W_{ba}$ is work done by the electric field. The function on the left is what called Electric potential. Note that only potential difference is defined not absolute value of potential.
The same can be done a charge say $q$, in this case $$U(\mathbf{r}_b)-U(\mathbf{r}_a)=-\int_{\mathbf{r}_a}^{\mathbf{r}_b}q\mathbf{E}\cdot d\mathbf{r}=-qW_{ba}$$ so that $$\Delta U = q\Delta V$$
Through the following you can deduce which option should be correct.
Q1. The potential difference or voltage $V$ between two points is defined as the work required per unit charge to move the charge between the two points, or $V=W/Q$.
When work is done to move change between two points there is a change in electrical potential energy of the charge.
Q2. Based on the definition of voltage, $\Delta V$ would mean the change in voltage or change in work required per unit charge to move the charge between the two points.
For example, let’s say a current of 1 ampere flows in a 1 Ohm resistor. Per Ohm’s law the voltage between the terminals of the resistor equals 1 volt. If I increase the current to 4 amperes the voltage will be 4 volts. By increasing the current the change in voltage is 3 volts, or $\Delta V$ = 3 volts.
Q3. Work equals the change in potential energy.
Hope this helps.
All your expressions are right if they are followed by appropriate definitions.
First: potential energy is always relative to some reference, and therefore never absolute. If you say write $V$ you would always have to define where $V=0$. Similarly, if you write $\Delta V$, you would always have to define between which to points. Usually, one put $V=0$ infinitely far from charges of this is possible. Second: Work is the energy you must provide to move a charge (or anythong else) a certain distance against an external force. In the special cases like in elektrostatics or gravity, where this external force is conservative you can define the potential energy as the work requirered to move a charge (or anythong else) to a certain position against the conservative force field. In other situations, like friction, which is not a conservative force, you cannot define a potential.
