The question of the equilibrium linear charge density on a charged straight conducting “wire” of finite length as its cross-sectional dimension becomes vanishingly small relative to the length is revisited in our didactic presentation. We first consider the wire as the limit of a prolate spheroidal conductor with semi-minor axis *a* and semi-major axis *c* when $a/c<<1.$ We then treat an azimuthally symmetric straight conductor of length $2c$ and variable radius $r(z)$ whose scale is defined by a parameter *a*. A procedure is developed to find the linear charge density $\lambda (z)$ as an expansion in powers of 1/Λ, where $\Lambda \u2261ln(4c2/a2),$ beginning with a uniform line charge density $\lambda 0.$ We show, for this rather general wire, that in the limit $\Lambda >>1$ the linear charge density becomes essentially uniform, but that the tiny nonuniformity (of order 1/Λ) is sufficient to produce a tangential electric field (of order $\Lambda 0)$ that cancels the zeroth-order field that naively seems to belie equilibrium. We specialize to a right circular cylinder and obtain the linear charge density explicitly, correct to order $1/\Lambda 2$ inclusive, and also the capacitance of a long isolated charged cylinder, a result anticipated in the published literature 37 years ago. The results for the cylinder are compared with published numerical computations. The second-order correction to the charge density is calculated numerically for a sampling of other shapes to show that the details of the distribution for finite 1/Λ vary with the shape, even though density becomes constant in the limit Λ→∞. We give a second method of finding the charge distribution on the cylinder, one that approximates the charge density by a finite polynomial in $z2$ and requires the solution of a coupled set of linear algebraic equations. Perhaps the most striking general observation is that the approach to uniformity as $a/c\u21920$ is extremely slow.

Skip Nav Destination

Article navigation

September 2000

September 01 2000

# Charge density on thin straight wire, revisited

J. D. Jackson

J. D. Jackson

University of California, Berkeley, California 94720

Search for other works by this author on:

*American Journal of Physics*68, 789–799 (2000)

Article history

Received:

November 12 1999

Accepted:

February 11 2000

Connected Content

A related article has been published:
Comment on “Charge density on a thin straight wire, revisited,” by J. D. Jackson [Am. J. Phys.

**68**(9), 789–799 (2000)]Citation

J. D. Jackson; Charge density on thin straight wire, revisited. * American Journal of Physics* 1 September 2000; 68 (9): 789–799. https://doi.org/10.1119/1.1302908

Download citation file:

## Sign in

Don't already have an account? Register

### Sign In

You could not be signed in. Please check your credentials and make sure you have an active account and try again.

Could not validate captcha. Please try again.

### Sign in via your Institution

Sign in via your InstitutionPay-Per-View Access

$40.00