How Work =Voltage * Charge? [closed]

Question

We know, Work/charge=volt So work =volt*charge But work =Fd So my question is since both voltage and charge are not dependable on distance,so how come work can be calculated without factoring in distance......

Answer 1

In regular mechanics the work done by a force moving from some start point $x_1$ to some end point $x_2$ is:

$$ W = \int_{x_1}^{x_2} F(x)~dx \tag{1} $$

We write this as an integral because the force may change as we move i.e. it may be a function of $x$. Now in an electric field $E$ the force the electric field exerts on a charge $q$ is:

$$ F=Eq $$

and just as before the work done is the integral of this force with distance:

$$ W = \int_{x_1}^{x_2} E(x)q~dx \tag{2} $$

This is the same as equation (1) just with the force replaced by $E(x)q$, where we write the field as a function of $x$ because like the force in equation (1) the field may change with position.

Now the charge $q$ is a constant so we can take it outside the integral to get:

$$ W = q~\int_{x_1}^{x_2} E(x)~dx \tag{3} $$

And the integral of the field with respect to distance has a special meaning. It is what we call the electric potential. So equation (3) is simply:

$$ W = q\left( V(x_2) - V(x_1) \right) = q\Delta V \tag{4} $$

where $\Delta V$ is the potential difference between $x_1$ and $x_2$.

And that's why the work is just $qV$. It's because $V$ is the integral wrt distance. We write $V$ rather than the integral because it's usually more convenient to do so.

You can pull the same trick with regular mechanics. For example when you move up or down a distance $h$ against gravity the work is:

$$ W = mgh $$

i.e. a force $mg$ times a distance $h$. But we can define a gravitational potential energy $U = gh$ and then write:

$$ W = mU $$

This is now analogous to our equation for the electrostatic work $W=qV$. In this simple case there probably doesn't seem to be much point in using a gravitational potential energy, but in more complicated calculations it's common to use gravitational potential energy rather than a force.

Answer 2

Voltage does depend on distance, it's dimensional formula is $ML^2/QT^2$

Answer 3

Work done is always against a force which may be contact force like friction or a non contact force such as magnetic or electrostatic. Such non contact forces are experienced in a field like electric field or gravitational field or magnetostatic field. In this case the fields is of electrostatic origin. Therefore in the field if different points have different field strengths they have different potentials. Therefore work needs to be done to move a charge against this force in the field. ( The work done may be positive or negative.) wish this clarifies :)