The geometric mean radius of a conductor, having four equal strands with each strand of radius ‘r’, as shown in the figure below, is

The correct answer is 1.723 r.

1. Understanding the Arrangement of Strands:
The given figure shows a conductor consisting of four identical, mutually touching circular strands, each with a physical radius of $r$.

Let us place the centers of these four strands at the vertices of a square:
- Let the strands be labeled 1, 2, 3, and 4 in clockwise order.
- The distance between the centers of any two adjacent touching strands (e.g., between 1 and 2, 2 and 4, 4 and 3, or 3 and 1) is equal to the sum of their radii:

$d_{12}=d_{13}=2r$

- The distance between the centers of diagonally opposite strands (e.g., between 1 and 4, or 2 and 3) is given by the diagonal of the square of side length $2r$:

$d_{14}=\sqrt{{\left(2r\right)}^2+{\left(2r\right)}^2}=2\sqrt{2}r$

2. Self-Geometric Mean Distance (GMR):
The self-GMD or Geometric Mean Radius ($D_s$) of a stranded conductor is calculated using the distances between all strands:

$D_s={\left[\prod _{i=1}^n\prod _{j=1}^n{d}_{ij}\right]}^{\frac{1}{n^2}}$

For $n=4$ identical symmetrically arranged strands, the product of distances from any single strand to all others (including its own self-GMR, $r\text{'}$) is identical. Therefore:

$D_s={\left[d_{11}\times d_{12}\times d_{13}\times d_{14}\right]}^{\frac{1}{4}}$

where:
- $d_{11}=r\text{'}=r\cdot e^{-1/4}\approx 0.7788r$ (the fictitious self-radius accounting for internal magnetic flux)
- $d_{12}=d_{13}=2r$
- $d_{14}=2\sqrt{2}r$

3. Step-by-Step Calculation:
Substituting these values into the expression for $D_s$:

$D_s={\left[0.7788r\times 2r\times 2r\times 2\sqrt{2}r\right]}^{\frac{1}{4}}$

$D_s={\left[0.7788\times 8\sqrt{2}\times r^4\right]}^{\frac{1}{4}}$

Since $\sqrt{2}\approx 1.4142$:

$0.7788\times 8\times 1.4142\approx 8.8111$

Taking the fourth root of the coefficient:

$D_s={\left(8.8111\right)}^{0.25}r\approx 1.7227r\approx 1.723r$

Thus, the geometric mean radius of the four-strand conductor is 1.723 r.