Electric Fields

Topic hub: This is the main overview page for Topic 15. Use it as the starting point, then follow the branch notes for deeper treatment of electric potential and energy, charged-particle motion, and common exam traps.

9749 syllabus focus

The examinable core is the field concept, field lines, Coulomb’s law, point-charge and uniform fields, force and motion in a uniform field, electric potential, potential gradient, and the electric–gravitational analogy. Equipotentials, energy changes and superposition connect these outcomes to earlier force and energy ideas.

Overview

This topic explains how charges exert forces at a distance and how the same physics can be described from both force and energy viewpoints.

A useful first-learner chain is

$$\text{source charge(s)}⟶\text{electric field at a point}⟶\text{force on a charge placed there}.$$

The source charge creates the field. A small positive test charge is an imagined probe used to define it without appreciably disturbing the source arrangement. Once the field is known, the force on any charge $q$ is $\vec{F}=q\vec{E}$.

The topic structure is:

• force between point charges
• electric field strength and direction
• field-line patterns and superposition
• electric potential and electric potential energy
• equipotentials and the potential-gradient relation
• charged-particle motion in uniform fields
• graph behaviour of $E$, $V$, and $U$

Core Ideas

Electric-fields questions revolve around a small linked set of ideas:

• charges exert forces on one another
• electric field strength is force per unit positive charge
• electric field is a vector quantity
• electric potential is a scalar quantity
• potential energy depends on both location and test charge
• uniform fields give constant force and therefore constant acceleration
• graph shape matters as much as formula use

Quantity	Meaning	Type	Depends on test charge?	SI unit
electric force $\vec{F}$	force on the charge placed in the field	vector	yes: $\vec{F}=q\vec{E}$	N
electric field $\vec{E}$	force per unit positive test charge	vector	no	N C${}^{-1}$ or V m${}^{-1}$
electric potential $V$	work done per unit positive charge from the reference point	scalar	no	V or J C${}^{-1}$
potential energy $U$	energy of the source–test-charge system	scalar	yes: $U=qV$	J

Exam Relevance

This topic is heavily tested through:

• direct substitution into Coulomb’s-law, field, potential, or energy formulas
• comparison of vector and scalar addition
• graph interpretation of $E$, $V$, and $U$ against distance
• charged-particle motion in uniform electric fields
• explanation questions involving direction, sign, and field-line reasoning

Coulomb’s Law

For two point charges separated by distance $r$, the magnitude of the force is:

$$F=\frac{1}{4\pi {\epsilon }_0}\frac{|Q_1{Q}_2|}{r^2}$$

where ${\epsilon }_0$ is the permittivity of free space. The equation applies to point charges in free space or air, and $r$ is their centre-to-centre separation.

It is important to recall that force is a vector. When two charges $Q$ and $q$ interact with each other, the force on $q$ exerted by $Q$ can be expressed as:

$$\vec{F}=\frac{1}{4\pi {\epsilon }_0}\frac{Qq}{r^2}\hat{r}$$

where $\hat{r}$ is a unit vector pointing from $Q$ to $q$.

• the force acts along the line joining the charges
• the charge signs determine whether the force is attractive or repulsive

Figure: The forces form a Newton’s-third-law pair: equal in magnitude, opposite in direction and collinear. Like charges repel and unlike charges attract. The separation $r$ is measured from centre to centre; the diagram does not use the arrow direction as a substitute for charge signs.

Meaning

• like charges repel
• unlike charges attract
• the magnitude falls as $1/r^2$

Electric force on a test charge due to a source charge acts along the line joining them; the charge signs determine attraction or repulsion.

Only after declaring a positive axis may Coulomb’s law be expressed as a signed one-dimensional component equation:

$$F=\frac{1}{4\pi {\epsilon }_0}\frac{Qq}{r^2}$$

Electric Field Strength

Electric field strength at a point is the force per unit positive charge. In vector form:

$$\vec{E}=\frac{\vec{F}}{q}$$

Here $q$ is a small positive test charge. “Small” means that it does not significantly rearrange the source charges. Units:

• N C${}^{-1}$
• V m${}^{-1}$

For the field generated by a point charge $Q$, the vector form is:

$$\vec{E}=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r^2}\hat{r}$$

where $\hat{r}$ is a unit vector in the out-pointing radial direction from the position of $Q$. The field direction is defined as the direction of force on a positive test charge:

• away from positive source charges
• towards negative source charges

Superposition

If several charges act at a point:

$${\vec{E}}_{\text{net}}={\vec{E}}_1+{\vec{E}}_2+{\vec{E}}_3+\cdots$$

This is a vector sum, so direction matters.

At the observation point, draw each field contribution, resolve into components, then add components. Do not add field magnitudes unless all contributions are collinear and a positive direction has been declared.

Figure: Electric-field contributions are vectors and must be added with direction. Electric potentials are scalars and are added algebraically with their signs. Therefore ${\vec{E}}_{\text{net}}=0$ does not necessarily imply $V_{\text{net}}=0$.

A revealing midpoint example

Two equal positive charges $+Q$ are separated symmetrically, and their midpoint is distance $a$ from each charge. At the midpoint, their field vectors have equal magnitudes and opposite directions:

$${\vec{E}}_{\text{net}}=\vec{0}.$$

Their potentials are both positive scalars:

$$V_{\text{net}}=\frac{kQ}{a}+\frac{kQ}{a}=\frac{2kQ}{a}>0.$$

This example prevents a common error: zero net field means zero local force per unit charge, not necessarily zero potential.

Electric Field Lines

Field lines are a visual model of the electric field.

Rules:

• leave positive charges and enter negative charges, or begin/end at infinity
• never cross
• the tangent gives the local field direction
• closer spacing represents a stronger field only as a qualitative convention within one consistently drawn diagram
• field lines are not particle trajectories

Common patterns:

• isolated positive charge: radially outward lines
• isolated negative charge: radially inward lines
• opposite charges: lines run from positive to negative
• like charges: lines curve away from the central region
• uniform field between parallel plates: parallel, evenly spaced lines

*Figure: Standard electric-field patterns. Arrowheads show the force direction on a positive test charge. Lines leave positive charge and enter negative charge (or extend to infinity), never cross, and are densest where the represented field is stronger. The parallel-plate panel describes only the approximately uniform central region.*

Field lines show direction and relative strength; equipotentials are perpendicular to the field direction.

Electric Potential and Electric Potential Energy

Electric potential at a point is the work done per unit positive charge in bringing a small test charge from infinity to that point, without changing its kinetic energy:

$$V=\frac{W_{\mathrm{ext}}}{q}.$$

Potential $V$ is a scalar property of the source field and the location. It exists even when no test charge is present. For an isolated charge distribution, $V=0$ at infinity is the conventional reference.

If a charge $q$ is placed at that location, the source–test-charge system has electric potential energy

$$U=qV.$$

Therefore $V$ is independent of the chosen test charge, whereas $U$ depends on both the location and the sign and magnitude of $q$. For a controlled movement with no kinetic-energy change, $W_{\mathrm{ext}}=\Delta U$; work done by the electric field is $W_{\mathrm{field}}=-\Delta U$.

Special Case: Field Due to a Point Charge $Q$

For a source point charge $Q$ in free space or air,

$$V=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r}.$$

The interaction energy when a test charge $q$ is placed there is

$$U=qV=\frac{1}{4\pi {\epsilon }_0}\frac{Qq}{r}.$$

Electric potential is a scalar quantity, so potentials from several source charges add algebraically:

$$V_{\text{net}}=V_1+V_2+V_3+\cdots$$

Sign Meaning

• like charges give positive potential energy
• unlike charges give negative potential energy
• this helps explain whether energy must be supplied or is released

Energy Change

For a charge moving between two points:

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

If only electric forces act:

$$\Delta K+\Delta U=0$$

See also Electric Potential and Energy.

Equipotentials and Potential Gradient

An equipotential surface is the set of points with the same electric potential.

Properties:

• no work is done moving a charge along an equipotential
• equipotentials meet field lines at right angles
• for equal potential intervals, closer spacing means a larger potential gradient and therefore a stronger field

Along a chosen $x$-direction, the component relation is:

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

For the magnitude of a uniform field between plates:

$$E=\frac{|\Delta V|}{d}$$

The field points toward decreasing potential. The syllabus statement that field strength is numerically equal to potential gradient refers to the equality of magnitudes.

Equipotential lines are perpendicular to electric field lines, and closer spacing between equipotentials indicates a stronger electric field when adjacent equipotentials represent equal potential intervals. For the electric field from a point charge, all points at the same distance from the charge have the same electric potential. Therefore, points of equal potential form concentric spherical surfaces, shown in cross-section as concentric circles.

Uniform Fields Between Parallel Plates

*Figure: Uniform electric field between oppositely charged parallel plates. The solid vertical arrows represent electric field lines directed from the positive plate to the negative plate. The dashed horizontal lines are equipotential lines, which are perpendicular to the electric field. Equal spacing of both field lines and equal-interval equipotentials indicates a uniform electric field with constant potential gradient.*

Figure: Potential changes linearly with perpendicular distance in the central uniform region. Moving in the field direction takes a positive test charge toward lower potential, so $E_x=-dV/dx$ and $E=|\Delta V|/d$.

Between oppositely charged parallel plates, the field in the central region is approximately uniform.

That means:

• constant field strength
• constant force on a charge
• constant acceleration for a charged particle

For a charge $q$:

$$\vec{F}=q\vec{E}$$

and:

$$\vec{a}=\frac{q\vec{E}}{m}$$

For one chosen component direction, the same relation may be written in signed-scalar form as $F=qE$ and $a=qE/m$.

Charged Particles in Electric Fields

Charged-particle motion can be treated using both energy and kinematics ideas.

Key cases

• a positive charge accelerates in the field direction
• a negative charge accelerates opposite the field direction
• a particle entering parallel to the plates keeps its horizontal velocity while accelerating vertically
• the path is parabolic

The parabolic-path statement assumes a uniform field perpendicular to the initial velocity, negligible fringing and other forces, and constant $q/m$. If weight may matter, compare $mg$ with $|q|E$ before neglecting it.

Typical relations

For a charged particle moving across a uniform field between two charged plates — the charged particle follows a parabolic path when horizontal motion combines with constant vertical acceleration.

$$t=\frac{L}{u_x}$$ $$y=\frac{1}{2}a{t}^2$$ $$v_y=at$$ $$\tan \theta =\frac{v_y}{v_x}$$ *Figure: Motion of an electron through a uniform electric field between oppositely charged parallel plates. Since the electron carries negative charge, its electric force $\vec{F}=-e\vec{E}$ and acceleration are opposite to the field direction. The electron therefore has constant vertical acceleration while maintaining horizontal motion, producing a parabolic trajectory inside the plates and a straight tangent path after leaving the field.*

Figure: With the same initial velocity and electric field, $+q$ and $-q$ have opposite force and acceleration directions. The field direction itself does not change when the test particle changes.

Graphs and Distance Trends

The main distance dependences are:

• $E\propto 1/r^2$
• $V\propto 1/r$
• $U\propto 1/r$ with sign set by the charge pair

The graph lesson is not just about formulas. It is also about recognising that the gradient of a potential graph gives field strength, and the negative gradient of a potential-energy graph gives the force component along that axis.

Figure: For a point charge, the field-strength magnitude falls as $1/r^2$, while electric potential and potential energy vary as $1/r$ with signs set by the source charge and test charge. The trends apply for $r>0$ and should not be extrapolated through the charge itself.

Comparison With Gravitational Fields

Electric and gravitational fields share:

• inverse-square force laws
• field strength as force per unit source quantity
• potential and potential-energy viewpoints
• superposition

Main differences:

• gravity acts on mass and is always attractive
• electric fields act on charge and can attract or repel
• electric potential can be positive or negative

Idea	Gravitational field	Electric field
source	mass $M$	charge $Q$
force on object	$\vec{F}=m\vec{g}$	$\vec{F}=q\vec{E}$
point-source field magnitude	$g=GM/r^2$	$E=
point-source potential	$\varphi =-GM/r$	$V=Q/(4\pi {\epsilon }_{0}r)$
interaction	always attractive	attractive or repulsive

Formula validity

Before substituting, ask whether the charge may be treated as a point charge, whether the field is uniform, whether $r$ is measured from the source, and whether the answer requires a magnitude, signed component or full vector.

See Gravitational vs Electric Fields.

Quick Revision Summary

• Coulomb’s law gives the force between point charges
• field strength is force per unit positive charge
• field lines show direction and strength
• potential is scalar; field is vector
• potential energy is $U=qV$
• equipotentials are perpendicular to field lines
• uniform fields give $E=|\Delta V|/d$ and constant acceleration
• charged particles in uniform fields follow parabolic paths if they enter sideways

Common Exam Traps

See Electric Fields Common Exam Traps for the compact checklist.

Links

• Electric Capacitance
• Electric Potential and Energy
• Charged Particles in Fields
• Electric Fields Common Exam Traps
• Gravitational vs Electric Fields
• Gravitational Fields