In the previous lesson, we described the interaction between charges using the concepts of force and the electric field. Here, we introduce an alternative, powerful approach using the concepts of energy and electric potential.
To apply conservation of energy, we need to define electric potential energy as we do for other types of potential energy. The electrostatic force between any two charges ($F=kQ_1Q_2/r^2$ ) is conservative since the dependence on position. Hence we can define potential energy $U$ for the electrostatic force.
The change in potential energy between two points, $a$ and $b$equals the negative of the work done by the conservative force as an object moves from $a$ to $b$:
$$ \Delta U=-W=U_b-U_a $$
consider the electric field between two plates, the field $\vec{\mathbf{E}}$ will be uniform over most of the region.
Now consider a tiny positive point charge $q$ placed at point a very near the positive plate. This charge $q$ is so small it has no effect on If this charge $q$ at point $a$ is released, the electric force will do work on the charge and accelerate it toward the negative plate. The work $W$ done by the electric field $E$ to move the charge a distance $d$ is
$$ W=Fd=qEd $$
The change in electric potential energy equals the negative of the work done by the electric force:
$$ U_b-U_a=-W=-qEd $$
In accord with the conservation of energy, electric potential energy is transformed into kinetic energy, and the total energy is conserved. Note that the positive charge $q$ has its greatest potential energy at point $a$, near the positive plate.
It is useful to define the electric potential as the electric potential energy per unit charge. Electric potential is given the symbol $V$. If a positive test charge $q$ in an electric field has electric potential energy at some point a, the electric potential at this point is
$$ V_a=\frac{U_a}{q} $$
Hence only the potential difference between two points $a$ and $b$ is measurable. When the electric force does positive work on a charge, the kinetic energy increases and the potential energy decreases. The potential difference is
$$ V_{ba}=\Delta V=V_b-V_a=\frac{U_b-U_a}{q}=\frac{-W_{ba}}{q} $$
The unit of electric potential, and of potential difference, is joules/coulomb and is given a special name, the volt, in honor of Alessandro Volta
If we wish to speak of the potential at some point $a$, we must be aware that depends on where the potential is chosen to be zero. The zero for electric potential in a given situation can be chosen arbitrarily, just as for potential energy, because only differences in potential energy can be measured. Often the ground, or a conductor connected directly to the ground (the Earth), is taken as zero potential, and other potentials are given with respect to ground.
In other cases, as we shall see, we may choose the potential to be zero at an infinite distance
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Suppose a negative charge, such as an electron, is placed near the negative plate at point $b$
If the electron is free to move, will its electric potential energy increase or decrease? How will the electric potential change?
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Because the electric potential difference is defined as the potential energy difference per unit charge, then the change in potential energy of a charge $q$ when moved between two points $a$ and $b$ is