Electric Field

The electric field $\vec{E}$ is a vector field that permeates all of space around electric charges. It represents the force per unit charge that a small positive test charge would experience at every point.

Coulomb's Law and the Electric Field

For a single point charge $q$ located at the origin, the field at position $\vec{r}$ is:

$\vec{E} = \frac{1}{4\pi\epsilon_0}\,\frac{q}{r^2}\,\hat{r}$

Symbol	Meaning
$\epsilon_0$	Permittivity of free space ($8.854 \times 10^{-12}$ F/m)
$q$	Source charge (positive or negative)
$r$	Distance from the charge to the field point
$\hat{r}$	Unit vector pointing from the charge to the field point

Superposition Principle

If multiple charges are present, the total field is the vector sum of the individual fields:

$\vec{E}_{\text{total}} = \sum_i \vec{E}_i$

This is why adding a second charge doesn't replace the first field — the arrows you see in the plot are the combined contribution from all charges.

Reading a Field-Line Diagram

• Direction of each arrow shows the direction of the force on a positive test charge.
• Density of arrows indicates field strength — closely packed arrows mean a stronger field.
• Field lines originate at positive charges and terminate at negative charges (or at infinity).

---

---

Things to Try

1. Single positive charge — field lines radiate outward uniformly in all directions, weakening with distance ($1/r^2$).
2. Dipole (one positive, one negative) — field lines arc from the positive charge to the negative charge. This is the most common configuration in nature (molecules, antennas, magnetic analogs).
3. Two positive charges — field lines repel between the charges, creating a "dead zone" (saddle point) at the midpoint where $\vec{E} = 0$.
4. Vary charge magnitudes — make one charge much larger than the other to see how the field becomes dominated by the stronger source.
5. Change separation — bring charges closer together to see the field intensify between them.

Where Electric Fields Appear

• Capacitors: Two parallel plates with opposite charges create a nearly uniform field between them — the basis of energy storage.
• Lightning: Charge separation in clouds creates enormous fields ($\sim 3 \times 10^6$ V/m) that ionize air, causing dielectric breakdown.
• Biological systems: The electric field across a cell membrane ($\sim 10^7$ V/m over ~10 nm) drives nerve impulses and ion transport.
• Particle accelerators: Carefully shaped electric fields accelerate charged particles to near light speed.

Limitations of the visualization — The quiver plot shows normalized arrows (all the same length) to keep the diagram readable. In reality, the field strength varies enormously — it's extremely strong near the charges and drops off as $1/r^2$. Also, this is a 2-D cross-section of what is really a 3-D field.

---

What Happens When Many Charges Dance in a Circle?

Arrange a ring of alternating positive and negative charges and watch the field lines weave between them. Increase the count to see how the pattern transforms from a simple multipole into something that resembles the field inside a real-world device. Try the dual-ring mode to see how two charge distributions interact across a gap.

Things to try

1. Start with 8 charges on a single ring -- you'll see a clear quadrupole pattern with field lines arcing between adjacent charges.
2. Crank it up to 32 charges -- the individual contributions blur into smooth, continuous field regions.
3. Switch to Dual Rings and widen the separation -- notice how the field between the rings weakens as the gap grows, and how each ring's field becomes increasingly self-contained.
4. Shrink the ring radius while keeping charge count high -- the field outside the ring becomes almost dipolar.