5.2.2. Electric Potential

Table of Contents

What electric potential means

Electric potential is a way to describe electric effects in space using energy per unit charge. Instead of focusing only on force, we ask a different question: if a small positive test charge were placed at some point, how much electric potential energy would it have per coulomb of charge?

The electric potential at a point is defined by

$$
V = \frac{U}{q}
$$

where $V$ is electric potential, $U$ is electric potential energy, and $q$ is the charge used to measure it.

This means potential tells us about the location itself, not about the specific test charge. A point in space can have a certain potential whether we place a charge there or not.

The SI unit of electric potential is the volt, written as $\text{V}$, where

$$
1\ \text{V} = 1\ \text{J/C}
$$

A volt is one joule of energy per coulomb of charge.

Electric potential is energy per unit charge.
$$
V = \frac{U}{q}
$$
Unit:
$$
1\ \text{V} = 1\ \text{J/C}
$$

Potential compared with potential energy

It is important not to confuse electric potential with electric potential energy. Potential energy depends on both the location and the charge placed there. Electric potential depends only on the location and the source charges creating the field.

If a point has potential $V$, then a charge $q$ placed there has potential energy

$$
U = qV
$$

A positive charge and a negative charge placed at the same point have different potential energies because their charges are different, even though the electric potential is the same.

This is similar to gravitational height. A hill has a certain height no matter what object you put there. But the gravitational potential energy depends on the mass of the object. In electricity, potential plays a role like height.

Electric potential is a property of position.
Electric potential energy depends on the charge placed there:
$$
U = qV
$$

Potential difference

In physics, we usually care more about differences in potential than about absolute potential. The potential difference between two points $A$ and $B$ is

$$
\Delta V = V_B - V_A
$$

This tells us how much the electric potential changes when moving from one point to another.

If a charge $q$ moves between these points, the change in electric potential energy is

$$
\Delta U = q \Delta V
$$

If $\Delta V$ is negative for a positive charge, then the charge loses electric potential energy. If $\Delta V$ is positive, then the charge gains electric potential energy.

Potential difference is often called voltage.

Work and electric potential

Electric potential is closely tied to work. When the electric force moves a charge, work is done and the charge's electric potential energy changes.

For a charge $q$ moving from point $A$ to point $B$,

$$
W_{\text{electric}} = -\Delta U
$$

Using $U = qV$, this becomes

$$
W_{\text{electric}} = -q \Delta V
$$

So if the electric force does positive work, the electric potential energy decreases.

We can also write potential difference as work per unit charge:

$$
\Delta V = -\frac{W_{\text{electric}}}{q}
$$

This is a very important idea. Voltage tells us how much energy change occurs for each coulomb of charge.

Potential difference is related to work by
$$
\Delta V = -\frac{W_{\text{electric}}}{q}
$$
and
$$
W_{\text{electric}} = -q\Delta V
$$

Sign of the potential

Electric potential can be positive, negative, or zero. The sign depends on the source charges and on where we choose the zero of potential.

A positive source charge tends to create positive potential around it. A negative source charge tends to create negative potential around it.

This does not mean that a positive potential is always "good" or a negative potential is always "bad". It only tells us the energy per unit positive charge relative to a chosen reference.

Potential due to a point charge

For a single point charge $Q$, the electric potential at distance $r$ from the charge is

$$
V = \frac{kQ}{r}
$$

where $k$ is Coulomb's constant.

This formula assumes that the potential is chosen to be zero infinitely far away.

Several important features follow from this formula. Potential is a scalar, so it adds algebraically. Also, the sign of $V$ is the sign of $Q$. As $r$ gets larger, the potential gets smaller in magnitude.

For a positive point charge, $V$ is positive everywhere. For a negative point charge, $V$ is negative everywhere.

Potential due to a point charge:
$$
V = \frac{kQ}{r}
$$
This is valid when
$$
V(\infty) = 0
$$

Superposition of potentials

If there are several point charges, the total electric potential at a point is the sum of the individual potentials:

$$
V_{\text{total}} = \sum_i \frac{kQ_i}{r_i}
$$

This is simpler than adding electric fields, because potential is a scalar quantity, not a vector.

A positive contribution raises the potential. A negative contribution lowers it.

A simple numerical example

Suppose a point charge $Q = 2.0 \times 10^{-6}\ \text{C}$ creates a potential at a point $0.30\ \text{m}$ away. Using

$$
V = \frac{kQ}{r}
$$

with $k = 9.0 \times 10^9\ \text{N m}^2/\text{C}^2$,

$$
V = \frac{(9.0 \times 10^9)(2.0 \times 10^{-6})}{0.30}
= 6.0 \times 10^4\ \text{V}
$$

So the point is at a potential of $60{,}000\ \text{V}$ relative to infinity.

If a charge $q = 3.0 \times 10^{-9}\ \text{C}$ is placed there, its potential energy is

$$
U = qV = (3.0 \times 10^{-9})(6.0 \times 10^4)
= 1.8 \times 10^{-4}\ \text{J}
$$

Positive and negative test charges

The meaning of potential becomes clearer when we think about different test charges.

If $V > 0$, then a positive charge has positive potential energy there, and a negative charge has negative potential energy there.

If $V < 0$, then a positive charge has negative potential energy there, and a negative charge has positive potential energy there.

The direction a charge tends to move depends on how its potential energy changes. Positive charges tend to move toward lower potential energy, and negative charges also tend to move toward lower potential energy, which can correspond to moving toward higher electric potential.

This is why the sign of the moving charge matters.

Common formulas summary

Quantity	Formula	Meaning
Electric potential	$V = U/q$	Energy per unit charge
Potential energy	$U = qV$	Energy of a charge at a point
Potential difference	$\Delta V = V_B - V_A$	Change in potential between two points
Work and voltage	$\Delta V = -W/q$	Work per unit charge
Point charge potential	$V = kQ/r$	Potential from a point charge

Visual idea of potential around a charge

For a positive charge, the potential is highest near the charge and decreases as we move away. For a negative charge, the potential is most negative near the charge and approaches zero far away.

Rendering scientific sketch…

Electric potential around a point charge

Why potential is useful

Electric potential is useful because it often makes problems easier. Since it is a scalar, we can add contributions directly without breaking them into components. It also connects naturally to energy, which is often easier to think about than force.

In later topics, electric potential will help describe circuits, capacitors, and charge distributions. For now, the main idea is simple: electric potential tells us how much electric potential energy each coulomb of charge would have at a location.

Electric potential is a scalar quantity, not a vector.
For many charges,
$$
V_{\text{total}} = \sum_i V_i
$$
This makes potential often easier to calculate than electric field.

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