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 option is 1.723 r.

Step-by-Step Explanation:

From the provided image, we observe four identical solid circular strands, each of radius r, arranged in a symmetrical square configuration such that each strand is touching its adjacent strands.

Let the centers of the four strands be labeled as 1, 2, 3, and 4 in a clockwise order.

1. Distance Between Strand Centers:
Since adjacent strands are touching, the distance between their centers is equal to twice the radius:
$$d_{12}=d_{23}=d_{34}=d_{41}=2r$$
The distance between diagonal strands (e.g., between strand 1 and strand 3) can be calculated using the Pythagorean theorem:
$$d_{13}=d_{24}=\sqrt{{\left(2r\right)}^2+{\left(2r\right)}^2}=\sqrt{8r^2}=2\sqrt{2}r\approx 2.828r$$

2. Self-GMR of a Single Strand:
The self-geometric mean radius (GMR) of a single solid circular conductor, accounting for internal flux linkage, is given by:
$$r^{\prime }=e^{-1/4}r\approx 0.7788r$$

3. Overall Geometric Mean Radius (GMR) of the Conductor:
Since all four strands are identical and symmetrically positioned, the overall GMR (denoted as $D_s$) is calculated as:
$$D_s={(r^{\prime }·d_{12}·d_{13}·d_{14})}^{1/4}$$
Substituting the distance values:
$$D_s={(0.7788r·2r·2\sqrt{2}r·2r)}^{1/4}$$
Simplifying the product inside the parentheses:
$$D_s={(0.7788·8\sqrt{2}·r^4)}^{1/4}$$
Substituting $\sqrt{2}\approx 1.4142$:
$$D_s\approx {(0.7788·11.3137·r^4)}^{1/4}$$
$$D_s\approx {\left(8.8111r^4\right)}^{1/4}$$
Calculating the fourth root:
$$D_s\approx 1.723r$$