Electric Field and Electric Potential

Page Claimed/Edited by Adhik Durga (Spring 2026)

Electric Field and Electric Potential

Electric fields and electric potential are two closely related ways of describing how electric charges interact. The electric field describes the force a charge would experience at a point, while electric potential describes the electric potential energy per unit charge at that point. These ideas are useful for understanding point charges, capacitors, circuits, conductors, and the motion of charged particles.

Electric Field

An electric field describes how a charge influences the space around it. If a positive test charge is placed in an electric field, it experiences an electric force. The electric field tells us both the strength and direction of that force.

The electric field is defined as

[math]\displaystyle{ \vec{E} = \frac{\vec{F}}{q} }[/math]

where [math]\displaystyle{ \vec{E} }[/math] is the electric field, [math]\displaystyle{ \vec{F} }[/math] is the electric force on the test charge, and [math]\displaystyle{ q }[/math] is the charge of the test particle.

This means that electric field represents force per unit charge. The units of electric field are

[math]\displaystyle{ \text{N/C} }[/math]

or equivalently

[math]\displaystyle{ \text{V/m} }[/math].

Direction of the Electric Field

The direction of the electric field is defined as the direction of the force on a positive test charge.

• Electric field lines point away from positive charges.
• Electric field lines point toward negative charges.
• A positive charge placed in an electric field accelerates in the direction of the field.
• A negative charge placed in an electric field accelerates opposite the direction of the field.

For a single positive point charge, field lines point outward in all directions. For a single negative point charge, field lines point inward from all directions.

Electric Field from a Point Charge

For a point charge, the magnitude of the electric field is

[math]\displaystyle{ E = \frac{k|Q|}{r^2} }[/math]

where [math]\displaystyle{ k = 8.99 \times 10^9 \ \text{N m}^2/\text{C}^2 }[/math], [math]\displaystyle{ Q }[/math] is the source charge, and [math]\displaystyle{ r }[/math] is the distance from the source charge.

This equation shows that electric field gets weaker as distance increases. Since the distance is squared, doubling the distance makes the field four times weaker.

Properties of Electric Field Lines

Electric field lines are a visual way to represent an electric field.

• Field lines start on positive charges and end on negative charges.
• The direction of the field line shows the direction of the force on a positive test charge.
• Closer field lines represent a stronger electric field.
• More spread-out field lines represent a weaker electric field.
• Electric field lines never cross because the electric field cannot point in two different directions at the same location.

Electric Potential

Electric potential, also called voltage, describes electric potential energy per unit charge. Unlike electric field, electric potential is a scalar quantity, meaning it has magnitude but no direction.

Electric potential is defined as

[math]\displaystyle{ V = \frac{U}{q} }[/math]

where [math]\displaystyle{ V }[/math] is electric potential, [math]\displaystyle{ U }[/math] is electric potential energy, and [math]\displaystyle{ q }[/math] is charge.

The unit of electric potential is the volt:

[math]\displaystyle{ 1 \ \text{V} = 1 \ \text{J/C} }[/math]

This means that one volt is equal to one joule of potential energy per coulomb of charge.

Electric Potential from a Point Charge

For a point charge, the electric potential is

[math]\displaystyle{ V = \frac{kQ}{r} }[/math]

Unlike electric field, electric potential can be positive or negative depending on the sign of the source charge. A positive charge creates positive electric potential, while a negative charge creates negative electric potential.

Electric potential is a scalar, so potentials from multiple charges can be added directly:

[math]\displaystyle{ V_{\text{total}} = V_1 + V_2 + V_3 + ... }[/math]

Electric field, however, is a vector, so electric fields must be added using direction.

Reference Point for Electric Potential

Electric potential is always measured relative to a chosen reference point. Common choices include:

• [math]\displaystyle{ V = 0 }[/math] at infinity for point charges
• [math]\displaystyle{ V = 0 }[/math] at the surface of a conductor
• [math]\displaystyle{ V = 0 }[/math] at the ground
• [math]\displaystyle{ V = 0 }[/math] at another convenient location

The exact value of electric potential depends on the reference point. However, potential difference is physically meaningful because it describes how much potential energy changes as a charge moves.

Electric Field vs. Electric Potential

A common mistake is confusing electric field with electric potential. They are related, but they are not the same quantity.

Quantity	Symbol	Type	Units	Meaning
Electric field	[math]\displaystyle{ \vec{E} }[/math]	Vector	N/C or V/m	Force per unit charge
Electric potential	[math]\displaystyle{ V }[/math]	Scalar	V	Potential energy per unit charge

Electric field has direction, while electric potential does not. Electric potential describes the amount of potential energy per charge at a point, while electric field describes how that potential changes with position.

Relationship Between Electric Field and Electric Potential

Electric field lines and equipotential lines for a positive and negative point charge. Field lines show the direction of the electric field, while equipotential lines show locations with the same electric potential. Image source: Wikimedia Commons.

Electric field and electric potential are closely connected. The electric field tells us how quickly the electric potential changes as we move through space.

In one dimension, the relationship is

[math]\displaystyle{ E_x = -\frac{\Delta V}{\Delta x} }[/math]

or, in calculus form,

[math]\displaystyle{ E_x = -\frac{dV}{dx} }[/math].

The negative sign means that the electric field points in the direction where electric potential decreases most rapidly. In other words, electric field points from higher electric potential toward lower electric potential.

For a uniform electric field, this can also be written as

[math]\displaystyle{ \Delta V = -E \Delta x }[/math].

This means that if you move in the direction of the electric field, electric potential decreases. If you move opposite the direction of the electric field, electric potential increases.

Intuitive Explanation

One way to understand electric potential is to compare it to height. Imagine electric potential as height on a hill and positive charges as objects on that hill. A positive charge naturally accelerates from higher electric potential toward lower electric potential, similar to how an object rolls downhill.

In this analogy:

• Electric potential is like height.
• Electric field is like slope.
• A stronger electric field means the electric potential changes more quickly with distance.
• A weaker electric field means the electric potential changes more slowly with distance.

This analogy is useful, but it is not perfect. A charge only moves according to the electric force if no other forces prevent it from moving.

Examples

Point Charge

For a positive point charge, the electric field points outward and the electric potential is positive. The electric potential is greatest near the charge and decreases as distance increases.

For a negative point charge, the electric field points inward and the electric potential is negative. As you move closer to the negative charge, the electric potential becomes more negative.

Uniform Electric Field

In a uniform electric field, the electric field has the same magnitude and direction everywhere. The field lines are equally spaced and parallel.

A common example is the region between two oppositely charged parallel plates. If the electric field points from the positive plate to the negative plate, then electric potential decreases in that direction.

Parallel-Plate Capacitor

Electric field and equipotential lines for a parallel-plate capacitor. The electric field is strongest between the plates, and the equipotential lines show regions of equal electric potential. Image source: Wikimedia Commons.

A parallel-plate capacitor consists of two conducting plates with opposite charges. Between the plates, the electric field is nearly uniform except near the edges.

• The electric field points from the positive plate to the negative plate.
• The positive plate is at a higher electric potential.
• The negative plate is at a lower electric potential.
• The electric potential changes almost linearly between the plates.

Worked Example

Suppose a uniform electric field has magnitude

[math]\displaystyle{ E = 200 \ \text{V/m} }[/math]

and points in the positive x-direction. How much does the electric potential change over a distance of

[math]\displaystyle{ \Delta x = 0.50 \ \text{m} }[/math]

in the direction of the field?

Using

[math]\displaystyle{ \Delta V = -E \Delta x }[/math],

we get

[math]\displaystyle{ \Delta V = -(200)(0.50) }[/math]

[math]\displaystyle{ \Delta V = -100 \ \text{V} }[/math].

The electric potential decreases by 100 V in the direction of the electric field.

Computational Model

The following GlowScript model shows the electric field around a positive and negative point charge. The arrows point away from the positive charge and toward the negative charge. The longer arrows represent stronger electric fields, which occur closer to the charges.

GlowScript 3.2 VPython

scene.title = "Electric Field Around Two Point Charges"

k = 9e9
scale = 2e-10

q1 = 1e-9
q2 = -1e-9

charge1 = sphere(pos=vector(-2,0,0), radius=0.25, color=color.red)
charge2 = sphere(pos=vector(2,0,0), radius=0.25, color=color.blue)

label(pos=charge1.pos + vector(0,0.5,0), text="+Q", height=16, box=False)
label(pos=charge2.pos + vector(0,0.5,0), text="-Q", height=16, box=False)

def electric_field(q, source_pos, point_pos):
    r = point_pos - source_pos
    if mag(r) == 0:
        return vector(0,0,0)
    return k*q*r/(mag(r)**3)

for x in range(-5,6):
    for y in range(-4,5):
        point = vector(x,y,0)
        if mag(point - charge1.pos) > 0.6 and mag(point - charge2.pos) > 0.6:
            E = electric_field(q1, charge1.pos, point) + electric_field(q2, charge2.pos, point)
            arrow(pos=point, axis=scale*E, color=color.orange, shaftwidth=0.04)

This model uses the point charge electric field equation. At each point on the grid, the program calculates the electric field from both charges and adds them together as vectors. This shows the principle of superposition, which means the total electric field is the vector sum of the fields created by each individual charge.

Common Mistakes

• Electric field is a vector, but electric potential is a scalar.
• Electric field lines point in the direction a positive test charge would accelerate.
• Electric potential does not have direction.
• A strong electric field means electric potential changes quickly with distance.
• Electric potential can be positive, negative, or zero depending on the source charges and reference point.
• Electric potential energy and electric potential are not the same thing. Electric potential energy depends on the charge placed in the field, while electric potential exists at a location regardless of what test charge is placed there.
• Electric field lines never cross because the field cannot have two directions at one point.

Summary

Electric field and electric potential are two different but connected ideas. Electric field describes force per unit charge and is a vector. Electric potential describes potential energy per unit charge and is a scalar. The electric field points in the direction where electric potential decreases most rapidly. Understanding both quantities makes it easier to analyze charges, capacitors, circuits, and energy changes in electric systems.

References

• OpenStax University Physics Volume 2, Chapter 5: Electric Charges and Fields
• OpenStax University Physics Volume 2, Chapter 7: Electric Potential
• Georgia Tech Physics 2212 course materials