Electric Field of a Point Charge

Equivalent forms

E = k_e \frac{q}{r^2}

\vec{E} = -\nabla V

The field concept transformed physics — instead of forces between distant objects, every charge shapes the space around it.

Symbol	Name	SI unit	Dimension	Range
$E$	Electric field magnitudeoutput Strength of the electric field at distance r from the charge	V/m	M·L·T⁻³·I⁻¹	0 – 10000000
$q$	Source charge Point charge creating the field	C	I·T	-0.00001 – 0.00001
$r$	Distance from charge Radial distance from the source charge	m	L	0.01 – 2

Unit systems

Symbol	SI	CGS	Imperial
$E$	V/m	statV/cm	V/m
$q$	C	statC	C
$r$	m	cm	ft

Where it holds

Valid for a single point charge in vacuum. For continuous distributions, integrate over the charge distribution. In dielectric media, the field is reduced by the relative permittivity

\varepsilon

ᵣ.

Dimensional analysis

$[E] = [k_e]\cdot [q]\cdot [r]^{-2} \to (N\cdot m^{2}\cdot C^{-2})(C)(m^{-2}) = N/C = V/m \checkmark$

Discovery

Michael Faraday · 1832

Faraday introduced the concept of 'lines of force' to visualize how electric influence propagates through space, replacing the action-at-a-distance view.

Try this

How does a charge 'know' another charge is nearby without touching it?

Find the electric field 0.3 m from a +4 μC point charge.

Research status: stable

Real-world applications

Electron beam deflection in CRT displays and electron microscopes
Electric field mapping in capacitor design
Lightning rod placement and electrostatic shielding
Particle accelerator beam steering

Common misconceptions

The field exists whether or not a test charge is present — it is a property of space
E points radially outward for positive charges, inward for negative
The field from a point charge is not uniform — it varies with distance

Experimental verification

Measured using electroscopes and later precision electrometers. Modern verification via Millikan-type experiments and atomic spectroscopy (Stark effect confirms the field's action on charges).

Derivation

Defined as F/q_test in the limit of vanishingly small test charge:

E = lim(q

_test

\to 0) F/q

_test

= k_e\cdot q/r^{2}

.

This definition ensures the test charge doesn't disturb the source distribution.

Limiting cases

r \to 0

⟶

E \to \infty

Field diverges at the location of the point charge — a singularity resolved by quantum theory.

r \to \infty

⟶

E \to 0

Field falls off as

1/r^{2}

, becoming negligible at large distances.

q \to 0

⟶

E \to 0

No source charge produces no field.

What if…

What if the charge is negative?

Field magnitude is the same but direction reverses — field lines point inward toward the negative charge.

What if you move to

10\times the

distance?

Field drops by $100\times$ . At $r = 3\,\mathrm{m}$ : $E = 3.99\times 10^{3} V/m$ . The $1/r^{2}$ falloff is dramatic.

What if you place a test charge in the field?

Force on test charge $q_{0} is F = q_{0}E$ . A +1 nC test charge at 0.3 m feels $F = 10^{-9} \times 3.99\times 10^{5} = 3.99\times 10^{-4} N$ .

1

Field near a point charge

Given ·

q:: 0.000004
r:: 0.3

Find · E

Steps

Identify: $q = +4 \mu C = 4\times 10^{-6} C$ , $r = 0.3\,\mathrm{m}$
Apply $E = k_e \times q / r^{2}$
Compute: $8.9875\times 10^{9} \times 4\times 10^{-6} = 35950$
Divide by $r^{2} = 0.09$ : $E = 35950 / 0.09 = 3.99\times 10^{5} V/m$
Direction: radially outward from the positive charge

Result ·

E = k_e \times q / r^{2} = 8.9875\times 10^{9} \times 4\times 10^{-6} / 0.09 = 3.99\times 10^{5} V/m

2

Superposition of two point charges

Given ·

q1:: 0.000002
q2:: -0.000002
d:: 0.4
point:: midpoint

Find · E at the midpoint

Steps

Midpoint is 0.2 m from each charge
E from +q points away from +q (to the right): $E_{1} = k_e \times 2\times 10^{-6} / 0.04 = 4.49\times 10^{5} V/m$
E from $-q$ points toward $-q$ (also to the right): $E_{2} = 4.49\times 10^{5} V/m$
Fields add (same direction): E_total $= 8.99\times 10^{5} V/m$
Direction: from the positive charge toward the negative charge

Result ·

E = 2 \times k_e \times q / (d/2)^{2} = 2 \times 8.9875\times 10^{9} \times 2\times 10^{-6} / 0.04 = 8.99\times 10^{5} V/m

(pointing from +

to -)

Coulomb's Law→Gauss's Law→Electric Potential (Point Charge)→