Two Charges and the Field Between Them

Every charge fills space with its own field, pointing away from a positive charge and toward a negative one, weakening with the square of the distance. With two charges the fields simply add as vectors, and that sum is the whole pattern: field lines bowing from + to −, or bending away where like charges push against each other.

How it works

Electric fields and superposition

In one sentence

The field of point charges $q_i$ at positions $\mathbf{x}_i$ is $\mathbf{E}(\mathbf{x}) = \sum_i \frac{k\,q_i}{|\mathbf{x}-\mathbf{x}_i|^2}\,\hat{\mathbf{u}}_i$ with $\hat{\mathbf{u}}_i = \frac{\mathbf{x}-\mathbf{x}_i}{|\mathbf{x}-\mathbf{x}_i|}$ . Superposition holds because Coulomb's law is linear in the charges.

General formula

{\htmlData{sym=0}{\mathbf{E}}} = \sum_i \frac{{\htmlData{sym=1}{k}}\,{\htmlData{sym=2}{q_i}}}{{\htmlData{sym=3}{d_i}}^{2}}\,{\htmlData{sym=4}{\hat{\mathbf{u}}_i}}, \qquad {\htmlData{sym=1}{k}} = 8.99 \times 10^{9}\ \text{N·m}^2/\text{C}^2

Each charge paints its own field over all of space. Add the arrows at every point and you get the field of the whole arrangement: no new physics, just vector addition.

What the symbols mean

$\mathbf{E}$
Electric field
The force per coulomb a small test charge would feel, a vector at every point.
$k$
Coulomb constant
8.99 × 10⁹ N·m²/C², which sets how strong electric forces are.
$q_i$
Charge i
The size and sign of each source charge.
$d_i$
Distance
How far the point is from charge i.
$\hat{\mathbf{u}}_i$
Direction
The unit vector pointing from charge i toward the point.

One charge $q$ makes a field of strength $kq/d^2$ at distance $d$ , pointing along $\hat{\mathbf{u}}$ , the unit vector from the charge to the point (reversed for a negative charge).
Coulomb's law is linear in the charges, so the field of several charges is the vector sum of their separate fields: superposition.
At each grid point the lab adds the two field vectors and draws the direction of the sum; the probe reports the magnitude in N/C (charges in nanocoulombs, distances in meters).
Between like charges the fields point opposite ways and cancel at the midpoint; between opposite charges they point the same way and add.

Learn the math

Electric Charge and Coulomb's LawE&M · Lesson 1, Charge and Electric Fields The Electric FieldE&M · Lesson 2, Charge and Electric Fields

Two Charges and the Field Between Them