Force in Electric Field Equation, Derivation & Rearrangements

Core idea

Overview

This equation defines the electrostatic force exerted on a point charge when placed within an external electric field. It illustrates that the force is directly proportional to both the magnitude of the charge and the intensity of the field, acting parallel to field lines for positive charges and anti-parallel for negative charges.

When to use: Use this equation when dealing with a point charge interacting with a known external electric field. It is applicable in scenarios involving uniform fields, such as those between parallel plates, or non-uniform fields where the field value at a specific point is known.

Why it matters: It is the fundamental principle behind particle accelerators, cathode ray tubes, and the operation of inkjet printers. Understanding this relationship allows engineers to precisely control the trajectory of charged particles in medical imaging and semiconductor manufacturing.

Symbols

Variables

F = Force, E = Field Strength, q = Charge

F

Force

N

E

Field Strength

N / C

q

Charge

C

Walkthrough

Derivation

Understanding Force in an Electric Field

Defines the fundamental relationship between a charged particle and the electric field it resides in.

The electric field is assumed to be uniform over the volume occupied by the particle.
The particle's own charge does not significantly alter the external field.

1

Define Electric Field Strength:

Electric field strength E is the force experienced per unit of positive charge.

E = \frac{F}{q}

2

Rearrange for Force:

Multiply both sides by q to find the total force acting on a charge in a given field.

F = q E

Note: In a uniform field between parallel plates, E is constant, so F is constant regardless of position.

Result

F = q E

Source: Edexcel A-Level Physics — Electric and Magnetic Fields

Free formulas

Rearrangements

Solve for $F$

Make F the subject

F = E q

F is already the subject of the formula.

Difficulty: 1/5

Solve for $E$

Rearrange Force in Electric Field for Electric Field Strength (E)

E = \frac{F}{q}

To make Electric Field Strength (E) the subject of the formula for Force (F) in an Electric Field, divide both sides by charge (q).

Difficulty: 2/5

Solve for $q$

Make q the subject

q = \frac{F}{E}

Start from Force in Electric Field. To make q the subject, divide by E.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a straight line passing through the origin because Force is directly proportional to Field Strength. For a student, this means that a small Field Strength results in a small Force, while a large Field Strength exerts a proportionally larger Force on the charge. The most important feature is that the linear relationship means doubling the Field Strength will always double the Force, provided the charge remains constant.

Graph type: linear

Why it behaves this way

Intuition

Imagine a charged particle as a tiny object being pushed or pulled by invisible 'field lines' that permeate space. The direction of the push/pull depends on the particle's charge relative to the field lines.

F

The electrostatic force exerted on a point charge.

It's the direct push or pull that causes a charged particle to accelerate or change its path.

E

The electric field strength at a specific point in space, representing the force per unit positive charge.

It describes the 'electrical environment' or 'influence' at a location, indicating how strongly a charge *would* be pushed or pulled there, regardless of whether a charge is actually present.

q

The magnitude and sign of the point charge.

It's the 'electrical quantity' of the particle; a larger magnitude means a stronger interaction with the field, and its sign determines the direction of the force relative to the field.

Signs and relationships

q: The sign of the charge 'q' determines the direction of the force 'F' relative to the electric field 'E'. A positive charge experiences a force in the same direction as 'E', while a negative charge experiences a force in

Free study cues

Insight

Canonical usage

This equation is typically used with SI units, where force is in Newtons, electric field strength in Newtons per Coulomb or Volts per meter, and charge in Coulombs.

Common confusion

A common mistake is failing to convert non-SI units (e.g., microcoulombs, millinewtons) into their base SI equivalents before calculation, or confusing the units for electric field strength (N/C vs V/m).

Unit systems

$F$ N · The Newton (N) is the SI unit of force.

$E$ N/C or V/m · Both Newtons per Coulomb (N/C) and Volts per meter (V/m) are equivalent SI units for electric field strength.

$q$ C · The Coulomb (C) is the SI unit of electric charge.

One free problem

Practice Problem

A proton with a charge of 1.6 × 10⁻¹⁹ C is placed in a uniform electric field with a strength of 450 N/C. What is the magnitude of the electric force acting on the proton?

Charge1.6e-19 C

Field Strength450 N/C

Solve for: $F$

Hint: Multiply the charge of the proton by the electric field strength.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Finding force on an ion between charged plates.

Study smarter

Tips

Ensure charge (q) is in Coulombs and field strength (E) is in N/C or V/m.
Remember that force is a vector; the direction is determined by the sign of the charge relative to the field direction.
For multiple fields, use the principle of superposition to find the net field (E) before calculating force.

Avoid these traps

Common Mistakes

Using E in V/m instead of N/C (they are equivalent).
Forgetting sign for negative charges.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Defines the fundamental relationship between a charged particle and the electric field it resides in.

Use this equation when dealing with a point charge interacting with a known external electric field. It is applicable in scenarios involving uniform fields, such as those between parallel plates, or non-uniform fields where the field value at a specific point is known.

It is the fundamental principle behind particle accelerators, cathode ray tubes, and the operation of inkjet printers. Understanding this relationship allows engineers to precisely control the trajectory of charged particles in medical imaging and semiconductor manufacturing.

Using E in V/m instead of N/C (they are equivalent). Forgetting sign for negative charges.