Electric Field Strength (Point charge) Equation, Derivation & Rearrangements

Core idea

Overview

This equation calculates the magnitude of the electric field intensity generated by a stationary point charge at a specific distance in a vacuum. It describes a fundamental inverse-square relationship where the strength of the field diminishes rapidly as one moves away from the source charge.

When to use: Use this formula when analyzing the electrostatic force per unit charge exerted by an isolated point charge or a spherically symmetric charge distribution. It is intended for use in vacuum or air, assuming the charge is stationary and localized at a single point in space.

Why it matters: Understanding point charge fields allows scientists to model the behavior of subatomic particles, design laboratory equipment like mass spectrometers, and calculate the forces involved in chemical bonding. It serves as the foundation for the principle of superposition, which is used to solve for complex electric fields in engineering.

Symbols

Variables

E = Electric Field Strength, Q = Point Charge, \varepsilon_0 = Permittivity of Free Space, r = Distance

E

Electric Field Strength

V / m

Q

Point Charge

C

ε_{0}

Permittivity of Free Space

F / m

r

Distance

m

Walkthrough

Derivation

Formula: Electric Field Strength (Point Charge)

States Coulomb's law for the electric field strength E = kQ/r² around a point charge Q.

Point charge or spherically symmetric charge distribution.
Medium is a vacuum (or permittivity ε₀ is used).

1

Start from Coulomb's Law:

The electrostatic force between charges Q and q separated by distance r.

F = \frac{1}{4 π ε _{0}} \frac{Qq}{r ^{2}}

2

Define Field Strength as Force per Unit Charge:

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

E = \frac{F}{q}

3

Obtain E for a Point Charge:

Dividing Coulomb's force by q gives the electric field due to charge Q.

E = \frac{1}{4 π ε _{0}} \frac{Q}{r ^{2}}

Result

E = \frac{1}{4 π ε _{0}} \frac{Q}{r ^{2}}

Source: AQA / OCR A-Level Physics — Electric Fields

Free formulas

Rearrangements

Solve for $E$

Make E the subject

E = \frac{Q}{4 π ε _{0} r ^{2}}

E is already the subject of the formula.

Difficulty: 1/5

Solve for $Q$

Make Q the subject

Q = E \cdot 4 π ε_{0} r^{2}

Rearranging the formula to solve for the point charge Q.

Difficulty: 2/5

Solve for $ε_{0}$

Make eps the subject

e p s = \frac{Q}{4 π E r ^{2}}

Rearranging the formula to solve for the permittivity of free space eps.

Difficulty: 3/5

Solve for $r$

Make r the subject

r = \frac{Q}{4 π ε _{0} E}

Rearranging the formula to solve for the distance r.

Difficulty: 3/5

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

Visual intuition

Graph

The graph of electric field strength (E) against distance (r) is a hyperbolic curve that approaches both axes as asymptotes. Because E is inversely proportional to the square of r, the curve drops steeply as distance increases, never reaching zero or touching the y-axis.

Graph type: hyperbolic

Why it behaves this way

Intuition

Imagine a point charge at the center of a sphere, emitting or absorbing electric field lines uniformly in all directions, with the density of these lines decreasing as the spherical surface area expands further from the

E

The magnitude of the force experienced per unit positive test charge at a specific point in space due to the source charge Q.

Represents how strong the electric 'push' or 'pull' would be on a tiny positive charge placed at that location. A stronger field means a stronger force.

Q

The magnitude of the source charge creating the electric field.

The 'strength' of the charge generating the field. A larger Q means a stronger field. Its sign determines the field's direction (outward for positive, inward for negative).

r

The radial distance from the point charge to the point where the electric field is being calculated.

How far away you are from the source charge. The field weakens rapidly as this distance increases.

ε_{0}

The permittivity of free space, a fundamental physical constant that quantifies the strength of the electric force between charges in a vacuum.

A measure of how easily an electric field can be established in a vacuum. It is part of the proportionality constant that relates charge and distance to field strength.

Signs and relationships

r^2 (in the denominator): The inverse-square dependence on distance arises because electric field lines spread out uniformly in three dimensions. The density of these lines, which represents field strength, decreases proportionally to the surface

Free study cues

Insight

Canonical usage

The equation is typically used with SI units, where electric field strength is expressed in Newtons per Coulomb or Volts per meter.

Common confusion

A common mistake is confusing the electric field strength (E) with electric potential (V) or using CGS (Gaussian) units without adjusting the formula, as the constant ε0 is specific to SI units.

Unit systems

$E$ N/C · Commonly expressed as Volts per meter (V/m), which is dimensionally equivalent to Newtons per Coulomb (N/C).

$Q$ C · Represents the magnitude of the point charge.

$r$ m · The radial distance from the point charge to the point where the electric field strength is being calculated.

$ε 0$ F/m · The permittivity of free space (vacuum permittivity), a fundamental physical constant.

One free problem

Practice Problem

Calculate the electric field strength E at a point 2.0 meters away from a point charge Q of 5.0 × 10⁻⁶ C in a vacuum.

Point Charge0.000005 C

Distance2 m

Permittivity of Free Space8.854e-12 F/m

Solve for: $E$

Hint: Square the distance in the denominator and use the value of pi as 3.14159.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Find E at 1mm from an electron (1.6e-19 C).

Study smarter

Tips

Ensure all units are converted to SI (meters for r, Coulombs for Q) before calculation.
The term 1/(4πeps) is approximately equal to 8.99 × 10⁹ N·m²/C² in a vacuum.
Remember that electric field strength is a vector; this formula provides the magnitude, while the direction depends on the charge's sign.
Because of the r² term, doubling your distance from a charge will result in the field strength becoming four times weaker.

Avoid these traps

Common Mistakes

Using distance completely instead of squared distance.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

States Coulomb's law for the electric field strength E = kQ/r² around a point charge Q.

Use this formula when analyzing the electrostatic force per unit charge exerted by an isolated point charge or a spherically symmetric charge distribution. It is intended for use in vacuum or air, assuming the charge is stationary and localized at a single point in space.

Understanding point charge fields allows scientists to model the behavior of subatomic particles, design laboratory equipment like mass spectrometers, and calculate the forces involved in chemical bonding. It serves as the foundation for the principle of superposition, which is used to solve for complex electric fields in engineering.

Using distance completely instead of squared distance.