Solution Found!
Consider an infinite chain of point charges, ±q (with
Chapter 2, Problem 33P(choose chapter or problem)
Consider an infinite chain of point charges, ±q (with alternating signs), strung out along the x axis, each a distance a from its nearest neighbors. Find the work per particle required to assemble this system. [Partial Answer: \(-a q^{2} /\left(4 \pi_{0} a\right)\), for some dimensionless number α; your problem is to determine α. It is known as the Madelung constant. Calculating the Madelung constant for 2- and 3-dimensional arrays is much more subtle and difficult.]
Questions & Answers
QUESTION:
Consider an infinite chain of point charges, ±q (with alternating signs), strung out along the x axis, each a distance a from its nearest neighbors. Find the work per particle required to assemble this system. [Partial Answer: \(-a q^{2} /\left(4 \pi_{0} a\right)\), for some dimensionless number α; your problem is to determine α. It is known as the Madelung constant. Calculating the Madelung constant for 2- and 3-dimensional arrays is much more subtle and difficult.]
ANSWER:
Step 1 of 3
In this problem we have to find the madelung constant for the infinite chain of point charges (alternative signs) and separated by a distance of
from each other.
The electrostatic potential energy at a distance from a charge
is,
---------------(1)
This is the work done on a test charge to bring it from infinity to the certain distance
from the source charge.
The work done per unit charge is defined as the electrostatic potential, which is,
---------------(2)