# Charged Rod

Surface · z = f(x, y, t)

`z = ∫₋₁¹ 1/hypot(x − u, y) du`

[Open in the app](https://www.wavelace.com/app#p=101) · [This page](https://www.wavelace.com/presets/charged-rod)

### What it draws

The height is a sum over a line of charge: `z = ∫₋₁^1 du / hypot(x − u, y)`. A point charge at `(u, 0)` has the Coulomb potential `1/distance`, in units that drop the constant, and `hypot(x − u, y)` is its distance from the point `(x, y)` of the sheet. Integrating `u` from −1 to 1 adds up a rod of length 2 along the x axis, one unit of charge per unit length. So the sheet is the electrostatic potential of a uniformly charged segment. There is no `t`: the field is static.

The integral has a closed form, `asinh((1 − x)/|y|) + asinh((1 + x)/|y|)`. Above the middle of the rod at `y = 1` it is `2 · asinh(1) ≈ 1.763`. Far away the rod looks like a point charge of 2 and `z ≈ 2/r`: at `(0, 3)` the sheet reads 0.655 against `2/3 ≈ 0.667`. Close in, `z ≈ 2 · ln(2/|y|)` grows without bound, 5.99 at `y = 0.1`. On the rod itself the integral is infinite. The grid line along `y = 0` only meets that at the origin, where a sample lands on the pole and the sheet has a hole. Elsewhere on the rod it reads 11 to 23, which the height clamp draws as a flat-topped wall.

### The shape

The level curves of the sheet are ellipses with their foci at the ends of the rod, `(−1, 0)` and `(1, 0)`. On the ellipse with semi-major axis `a` the potential is `ln((a + 1)/(a − 1))` , `ln 3 ≈ 1.099` on the one through `(2, 0)`. Near the rod the contours hug it; far out they round into circles. The rod is the degenerate ellipse in the middle, the ridge line. Its ends stand lower than its middle, 3.69 at `(1, 0.1)` against 5.99 above the centre, since charge lies on one side only.

### History

Charles-Augustin de Coulomb measured the inverse-square force between charges with a torsion balance in 1785. The potential, one number at each point whose slope is the field, took its name from George Green's essay of 1828.

### Try

- `Top`: the contours are the confocal ellipses, thin round the rod and nearly circular far out.
- Lengthen the rod, `integral(1/hypot(x − u, y), u, −2, 2)`: twice the charge, so the far field is `4/r` and the foci move to `(±2, 0)`.
- Set `Mesh` to an odd count such as 45: no grid line runs along the rod, the wall and the hole go, and the ridge tops out at the nearest rows, 6.8 at `y ≈ ±0.067`.
- Bend it into a ring, `integral(1/hypot(x − cos(u), y − sin(u)), u, 0, 2·π)`: a charged circle of radius 1, the centre its lowest point inside at `2π ≈ 6.28`, ringed by a circular wall.

### Read more

- [Electric potential](https://en.wikipedia.org/wiki/Electric_potential)
- [Coulomb's law](https://en.wikipedia.org/wiki/Coulomb%27s_law)
- [Equipotential](https://en.wikipedia.org/wiki/Equipotential)
- [George Green (mathematician)](https://en.wikipedia.org/wiki/George_Green_(mathematician))