KAHIBARO
Table of Contents
Meaning of the Electric Field
The electric field is a way to describe how electric charges influence the space around them. Instead of saying that one charge somehow reaches across empty space and pulls or pushes another charge directly, physics describes this influence by assigning a field to every point in space.
If a small positive test charge is placed at some point, the electric field tells us what electric force that test charge would feel there. This makes the electric field a property of space around charges, not just a property of the test charge itself.
The electric field at a point is defined as
$$
\vec{E} = \frac{\vec{F}}{q}
$$
where $\vec{F}$ is the electric force on a test charge $q$ placed at that point.
Here, $\vec{E}$ is a vector, so it has both magnitude and direction. Its direction is the direction of the force on a positive test charge.
Electric Field as Force per Unit Charge
Suppose a test charge $q$ is placed in a region where electric effects are present. If it experiences a force $\vec{F}$, then the electric field at that location is the force divided by the charge.
Rearranging the definition gives the force on a charge in a known field:
The electric force on a charge in an electric field is
$$
\vec{F} = q\vec{E}
$$
A positive charge feels force in the same direction as $\vec{E}$.
A negative charge feels force in the opposite direction.
This is one of the most important relations in electricity. It shows that the field exists independently, and any charge placed there responds to it according to its own value and sign.
Direction of the Electric Field
Because the electric field is defined using a positive test charge, its direction is always the direction that a positive charge would move if released.
This leads to a simple rule near isolated charges. Around a positive source charge, the field points away from the charge. Around a negative source charge, the field points toward the charge.
Electric Field of a Point Charge
For a single point charge $Q$, the electric field at a distance $r$ from the charge comes from Coulomb's law. Its magnitude is
$$
E = k\frac{|Q|}{r^2}
$$
where $k$ is Coulomb's constant.
In vector form, the direction is radial, meaning along the line joining the source charge and the point where the field is measured. If $Q$ is positive, the field points outward. If $Q$ is negative, the field points inward.
For a point charge $Q$, the magnitude of the electric field is
$$
E = k\frac{|Q|}{r^2}
$$
This is an inverse square law, so doubling the distance makes the field four times smaller.
This shows that the field becomes weaker as you move farther from the charge.
Units of Electric Field
From the definition $\vec{E} = \vec{F}/q$, the SI unit of electric field is newton per coulomb, written as $\text{N/C}$.
Using ideas from electric potential, another common unit is volt per meter, written as $\text{V/m}$. These units are equivalent, though that relationship is developed more fully in the chapter on electric potential.
$$
1\ \text{N/C} = 1\ \text{V/m}
$$
Field Strength and Distance
The strength of the electric field created by a point charge depends on two main things, the amount of charge and the distance from it. A larger source charge creates a stronger field, and greater distance gives a weaker field.
The table below summarizes this behavior.
| Quantity changed | Effect on field magnitude | ||
|---|---|---|---|
| Increase source charge $ | Q | $ | Field increases |
| Decrease source charge $ | Q | $ | Field decreases |
| Increase distance $r$ | Field decreases as $1/r^2$ | ||
| Decrease distance $r$ | Field increases as $1/r^2$ |
Example of Field and Force
Suppose a charge $q = 2.0 \times 10^{-6}\ \text{C}$ is placed in an electric field of magnitude $E = 300\ \text{N/C}$.
The force magnitude is
$$
F = qE = (2.0 \times 10^{-6})(300) = 6.0 \times 10^{-4}\ \text{N}
$$
If the charge is positive, the force is in the direction of the field. If the charge were negative, the force would have the same magnitude but opposite direction.
Electric Field as a Vector Quantity
The electric field is a vector, so fields from different charges combine by vector addition. This means that if several charges are present, the total field at a point is the vector sum of the fields produced by each charge separately.
The detailed method of combining fields belongs with superposition, but it is important here to remember that electric fields do not simply add as ordinary numbers unless they lie along the same line.
For example, two equal fields in opposite directions can cancel, giving zero net field at some point.
Visualizing the Field Around a Charge
The field near a point charge is symmetric. At the same distance from the charge, the field has the same magnitude in every direction. Only the direction changes, always pointing radially inward or outward.
The dashed circles represent points at equal distance from the charge. Since the distance is the same everywhere on each circle, the field magnitude is the same at all those points.
Important Ideas to Remember
The electric field is one of the central ideas in electromagnetism. It lets us describe how charges affect the space around them and how other charges respond when placed there.
Key facts about electric field:
$$
\vec{E} = \frac{\vec{F}}{q}
$$
$$
\vec{F} = q\vec{E}
$$
For a point charge,
$$
E = k\frac{|Q|}{r^2}
$$
The field direction is defined as the direction of force on a positive test charge.
Understanding electric field prepares you for field lines, dipoles, flux, and Gauss's law, where the geometry and behavior of fields become even more important.