1. Use the scaling of self-inductance

For a coil, self-inductance scales as

$L \propto \mu \frac{N^2 A}{\ell}$

where:

• $N$ = total number of turns,
• $A$ = area enclosed,
• $\ell$ = magnetic path length / linear size scaling term.

For similar shapes enlarged uniformly by a factor $k$:

• linear dimensions $\to k$ times,
• area $A \to k^2 A$,
• perimeter/length $\ell \to k\ell$.

2. Effect on number of turns

It is given that the number of turns per unit length of the frame remains same.

Since the perimeter of the equilateral triangular frame increases by factor $3$, the total number of turns also increases by factor $3$:

$N' = 3N$

3. Effect on area and length

If each side becomes $3$ times,

• enclosed area of the equilateral triangle becomes

$A' = 3^2 A = 9A$

• characteristic length/perimeter becomes

$\ell' = 3\ell$

4. New inductance

Now,

$L' \propto \frac{{N'}^2 A'}{\ell'}$

So,

$\frac{L'}{L} = \frac{(3N)^2 \cdot 9A / (3\ell)}{N^2 A / \ell}$

$\frac{L'}{L} = \frac{9 \cdot 9}{3} = 27$

Thus,

$L' = 27L$

5. Check options

• A: decreases by $9\sqrt{3}$ — incorrect
• B: increases by $27$ — correct
• C: decreases by $9$ — incorrect
• D: increases by $3$ — incorrect

Therefore, the self-inductance increases by a factor of 27.