PhysicsElectric FieldsA-Level

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Electric Field (Point Charge)

Field strength due to a point charge.

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E = \frac{Q}{4 π ϵ _{0} r ^{2}}

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Core idea

Overview

This equation defines the magnitude of the electric field produced by a stationary point charge in a vacuum. It describes how the field strength is directly proportional to the quantity of charge and decreases according to the inverse square of the distance from that charge.

When to use: Apply this formula when calculating the field intensity at a specific point in space surrounding a single, isolated charged particle. It assumes the charge is concentrated at a single point and that the surrounding environment is a vacuum or air, where the permittivity of free space is applicable.

Why it matters: This principle is the cornerstone of electrostatics, explaining how particles exert forces on each other without direct contact. It is essential for engineering electronic sensors, managing high-voltage insulation, and understanding the behavior of subatomic particles in physics research.

Symbols

Variables

E = Field Strength, Q = Charge, \epsilon_0 = Permittivity, r = Distance

E

Field Strength

N / C

Q

Charge

C

ϵ_{0}

Permittivity

F / m

r

Distance

m

Walkthrough

Derivation

Derivation of Electric Field (Point Charge)

Calculates the electric field strength at a distance from a point charge, derived from Coulomb's Law.

The source charge is a point charge or a uniform spherical charge.
The test charge is small enough not to distort the field.

Start with Coulomb's Law:

This is the force between a source charge Q and a test charge q separated by distance r.

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

Apply the Definition of Electric Field:

Electric field strength is defined as the force per unit positive charge.

E = \frac{F}{q}

Substitute and Simplify:

The test charge q cancels out, leaving the field strength dependent only on the source charge Q and the distance squared.

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

Result

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

Source: OCR A-Level Physics A — 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 = 4 π ϵ_{0} r^{2} E

To make $Q$ the subject of the Electric Field (Point Charge) formula, multiply both sides by $4 π ϵ_{0} r^{2}$ to isolate $Q$ on one side of the equation.

Difficulty: 2/5

Solve for $ϵ_{0}$

Make epsilon0 the subject

ϵ_{0} = \frac{Q}{4 π E r ^{2}}

Rearrange the Electric Field (Point Charge) formula to make $ϵ_{0}$ (permittivity of free space) the subject. This involves multiplying by the denominator, grouping the remaining terms, and dividing to isolate $ϵ_{0}$ .

Difficulty: 2/5

Solve for $r$

Make r the subject

r = \frac{Q}{4 π ϵ _{0} E}

To make r the subject from the electric field strength equation, first clear the denominator, then isolate r², and finally take the square root.

Difficulty: 2/5

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

Visual intuition

Graph

The graph displays an inverse-square relationship where field strength drops rapidly as distance increases, restricted to positive values with the axes acting as asymptotes. For a student, this means that very small distances result in an extremely intense field, while large distances cause the field strength to weaken significantly. The most important feature is that the curve never reaches zero, meaning the influence of a point charge theoretically extends across all space regardless of how far away you move.

Graph type: power_law

Why it behaves this way

Intuition

Imagine a point charge as a central source from which electric field lines emanate uniformly in all directions, spreading out over increasingly larger spherical surfaces, causing the field's intensity to diminish rapidly

Magnitude of the electric field strength at a point

Represents the force that a unit positive test charge would experience at that location. A stronger field means a greater force.

Magnitude of the point charge creating the electric field

The 'source' of the field. A larger charge produces a proportionally stronger field everywhere around it.

Radial distance from the point charge to the point where the field is measured

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

ϵ_{0}

Permittivity of free space (vacuum permittivity)

A fundamental constant that describes how easily an electric field can be established in a vacuum. It influences the strength of the field for a given charge and distance.

Signs and relationships

r^2 in the denominator: The inverse square dependence (1/ $r^{2}$ ) arises because electric field lines from a point charge spread out uniformly in three dimensions.
The overall positive value of E: This formula calculates the magnitude of the electric field strength, which is always a positive scalar quantity. The direction of the electric field (radially outward or inward)

Free study cues

Insight

Canonical usage

This equation is almost universally applied using SI units for all quantities.

Common confusion

Forgetting to convert charge (Q) from non-SI units (e.g., microcoulombs) to Coulombs (C) or distance (r) from centimeters to meters (m) before calculation.

Unit systems

$E$ N C^-1 · Also commonly expressed as V m^-1, which is dimensionally equivalent.

$Q$ C · Charge must be in Coulombs.

$r$ m · Distance must be in meters.

$e p s i l o n_{0}$ F m^-1 · Permittivity of free space, a fundamental physical constant.

One free problem

Practice Problem

Calculate the magnitude of the electric field at a point 3.0 meters away from a point charge of 1.0 microCoulomb in a vacuum.

Charge0.000001 C

Distance3 m

Permittivity8.854e-12 F/m

Solve for: $E$

Hint: Convert 1.0 microCoulomb to 10⁻⁶ Coulombs and calculate r² as 9.0.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Estimating field strength near a charged sphere.

Study smarter

Tips

Ensure the distance r is measured in meters and squared in the denominator.
The constant 1/(4πε₀) is approximately 8.99 × 10⁹ N·m²/C², which can simplify mental checks.
The electric field is a vector; this formula provides the magnitude, while the charge sign determines direction.
Always convert micro-Coulombs (µC) or nano-Coulombs (nC) to standard Coulombs before calculation.

Avoid these traps

Common Mistakes

Using r instead of r².
Mixing microcoulombs and coulombs.

Common questions

Frequently Asked Questions

Calculates the electric field strength at a distance from a point charge, derived from Coulomb's Law.

Apply this formula when calculating the field intensity at a specific point in space surrounding a single, isolated charged particle. It assumes the charge is concentrated at a single point and that the surrounding environment is a vacuum or air, where the permittivity of free space is applicable.

This principle is the cornerstone of electrostatics, explaining how particles exert forces on each other without direct contact. It is essential for engineering electronic sensors, managing high-voltage insulation, and understanding the behavior of subatomic particles in physics research.

Using r instead of r². Mixing microcoulombs and coulombs.

Estimating field strength near a charged sphere.

Ensure the distance r is measured in meters and squared in the denominator. The constant 1/(4πε₀) is approximately 8.99 × 10⁹ N·m²/C², which can simplify mental checks. The electric field is a vector; this formula provides the magnitude, while the charge sign determines direction. Always convert micro-Coulombs (µC) or nano-Coulombs (nC) to standard Coulombs before calculation.

References

Sources

Halliday, Resnick, and Walker, Fundamentals of Physics, 10th ed.
Griffiths, David J. Introduction to Electrodynamics, 4th ed.
Wikipedia: Electric field
NIST CODATA: Permittivity of vacuum
NIST CODATA
Halliday, Resnick, and Walker, Fundamentals of Physics
Halliday, Resnick, and Walker Fundamentals of Physics
Wikipedia article Electric field

Electric Field (Point Charge)

Overview

Variables

Derivation

Start with Coulomb's Law:

Apply the Definition of Electric Field:

Substitute and Simplify:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Force in Electric Field

Frequently Asked Questions

Sources