Physics 5. Part IV: Electricity and Magnetism 5.2. Electric Potential 5.2.4. Relationship Between Potential and Field

5.2.4. Relationship Between Potential and Field

Table of Contents

From Potential Difference to Electric Field

Electric potential and electric field describe the same electric situation in two closely related ways. The electric field tells us how strongly a positive test charge is pushed and in what direction. Electric potential tells us how much electric potential energy there is per unit charge at a point. The relationship between them is one of the central ideas in electrostatics.

If the potential changes from one place to another, there must be an electric field in that region. If the potential is the same everywhere in a region, then the electric field there is zero.

A spatial change in electric potential produces an electric field.
If the potential does not change with position, then
$$\vec{E} = \vec{0}.$$

Potential Difference and Work

When a charge moves in an electric field, the field can do work on it. The potential difference between two points is related to the work done by the electric force.

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

$$\Delta V = V_B - V_A = -\frac{W_{A \to B}}{q}.$$

The minus sign is important. It means that if the electric field does positive work on a positive charge, the electric potential decreases.

This gives a physical interpretation. Positive charges naturally move from higher electric potential to lower electric potential, if only the electric force acts. Negative charges behave oppositely because their charge is negative.

For a positive test charge, the electric field points in the direction of decreasing potential.

Field as the Rate of Change of Potential

The electric field is related to how fast the potential changes with position. In one dimension, along the $x$ axis, the relationship is

$$E_x = -\frac{dV}{dx}.$$

This means the $x$ component of the electric field equals the negative slope of the potential graph.

If $V(x)$ decreases rapidly as $x$ increases, then $E_x$ is large and positive. If $V(x)$ increases as $x$ increases, then $E_x$ is negative. If the graph is flat, then the field is zero.

In full vector form, the relationship is

$$\vec{E} = -\nabla V.$$

This expression says that the electric field points in the direction where the potential decreases most quickly.

Key formula:
$$\vec{E} = -\nabla V$$
In one dimension:
$$E_x = -\frac{dV}{dx}$$

Meaning of the Negative Sign

The negative sign often causes confusion, but its meaning is simple. The electric field points downhill in potential.

This is similar to how an object on a hill tends to move downhill, not uphill. In the same way, a positive charge tends to move toward lower electric potential.

A stronger field means a steeper drop in potential. A weaker field means a gentler change in potential.

Uniform Electric Field

A very important special case is a uniform electric field, such as the ideal field between two large parallel plates. If the field is constant and points along the $x$ direction, then

$$E_x = -\frac{\Delta V}{\Delta x}.$$

So the potential changes linearly with position:

$$V(x) = V_0 - E_x x,$$

if $V_0$ is the potential at $x = 0$.

This means that in a uniform electric field, equal distances correspond to equal changes in potential.

Situation	Potential behavior
Strong uniform field	Potential changes rapidly with distance
Weak uniform field	Potential changes slowly with distance
Zero field	Potential is constant

Example of Direction

Suppose the potential is higher on the left and lower on the right. Then the electric field points from left to right.

If instead the potential increases to the right, then the electric field points to the left.

So when reading a potential graph, do not look for where the potential is largest. Look for how the potential changes with position.

Potential Graphs in One Dimension

A graph of $V$ versus $x$ gives direct information about the electric field.

If the graph slopes downward, then $E_x$ is positive.
If the graph slopes upward, then $E_x$ is negative.
If the graph is flat, then $E_x = 0$.
If the graph is steep, the field is strong.

Shape of $V(x)$ graph	Sign of $E_x$	Field strength
Sloping downward with $x$	Positive	Depends on steepness
Sloping upward with $x$	Negative	Depends on steepness
Horizontal	Zero	Zero
Steeper slope	Same sign as slope rule	Larger magnitude

Equipotential Surfaces and Field Direction

Electric field lines are always perpendicular to equipotential surfaces. This follows from the fact that moving along an equipotential surface does not change the potential, so the electric field has no component along that surface.

If there were a component of electric field tangent to the equipotential surface, a charge moving a small distance along that surface would experience a change in potential, which would contradict the definition of equipotential.

Electric field lines are perpendicular to equipotential surfaces.
A charge moving along an equipotential surface experiences no change in electric potential:
$$\Delta V = 0.$$

Differential Form of Potential Difference

For a very small displacement $d\vec{r}$, the change in potential is

$$dV = -\vec{E} \cdot d\vec{r}.$$

This is a very useful formula. It says that only the component of displacement along the field affects the potential. If the displacement is perpendicular to the field, then $\vec{E} \cdot d\vec{r} = 0$, so the potential does not change.

This is another way to understand equipotential surfaces. Moving along them means moving perpendicular to $\vec{E}$.

Recovering Potential from the Field

If the electric field is known, the potential difference between two points can be found by integrating:

$$V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{r}.$$

This equation is the reverse of $\vec{E} = -\nabla V$. One tells us how to get field from potential, the other tells us how to get potential difference from field.

In electrostatics, the result depends only on the starting and ending points, not on the path taken between them.

Potential difference from the electric field:
$$V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{r}$$

Visual Picture

The connection between potential and field can be pictured as a landscape. Potential is like height on the landscape. The electric field is like the steepest downhill direction. Steeper regions correspond to stronger fields, and flat regions correspond to zero field.

Rendering scientific sketch…

Potential decreasing in the direction of the electric field

Two-Dimensional View

In two dimensions, the electric field has components in both directions:

$$\vec{E} = E_x \hat{i} + E_y \hat{j}.$$

The potential determines these components through

$$E_x = -\frac{\partial V}{\partial x}, \qquad E_y = -\frac{\partial V}{\partial y}.$$

So each component of the field is the negative rate of change of potential in that direction.

If the potential changes only with $x$, then there is no $y$ component of the field. If the potential changes only with $y$, then there is no $x$ component.

Equipotentials and Field Lines

A useful visual rule is that field lines cross equipotential lines at right angles. Where equipotential lines are close together, the potential changes rapidly with distance, so the electric field is strong. Where they are far apart, the field is weak.

Rendering scientific sketch…

Equipotential lines and electric field lines

Summary of the Relationship

Electric potential and electric field are two descriptions of the same electric reality. Potential is a scalar quantity, while electric field is a vector quantity. The electric field points toward lower potential, and its magnitude tells how quickly the potential changes with distance.

Essential relationships:
$$dV = -\vec{E} \cdot d\vec{r}$$
$$V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{r}$$
$$\vec{E} = -\nabla V$$
For one dimension:
$$E_x = -\frac{dV}{dx}$$

Understanding this relationship makes it possible to move back and forth between field descriptions and potential descriptions, which is one of the most powerful tools in electricity.

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