If $$K_1$$ and $$K_2$$ are the thermal conductivities, $$L_1$$ and $$L_2$$ are the lengths and $$A_1$$ and $$A_2$$ are the cross sectional areas of steel and copper rods respectively such that $$\dfrac{K_2}{K_1} = 9$$, $$\dfrac{A_1}{A_2} = 2$$, $$\dfrac{L_1}{L_2} = 2$$. Then, for the arrangement as shown in the figure, the value of temperature $$T$$ of the steel-copper junction in the steady state will be

We need to find the steady-state junction temperature T between a steel rod and a copper rod connected in series, with the steel end at 0°C and the copper end at 50°C.

Since the system is in steady state, the rate of heat flow through both rods is equal, so $$\frac{K_1 A_1 (T - 0)}{L_1} = \frac{K_2 A_2 (50 - T)}{L_2}$$ where subscript 1 refers to steel and subscript 2 refers to copper.

We are given that $$\frac{K_2}{K_1} = 9$$, $$\frac{A_1}{A_2} = 2$$, and $$\frac{L_1}{L_2} = 2$$.

Substituting these ratios into the heat balance equation gives $$\frac{K_1 A_1}{L_1} \cdot T = \frac{K_2 A_2}{L_2} \cdot (50 - T)\,.$$

From the above, the ratio of the coefficients is $$\frac{K_1 A_1 / L_1}{K_2 A_2 / L_2} = \frac{K_1}{K_2} \cdot \frac{A_1}{A_2} \cdot \frac{L_2}{L_1} = \frac{1}{9} \times 2 \times \frac{1}{2} = \frac{1}{9}\,.$$

Substituting this result into the previous relation yields $$\frac{1}{9} \cdot T = (50 - T)\,, $$ which can be written as $$T = 9(50 - T) = 450 - 9T\,, $$ leading to $$10T = 450$$ and hence $$T = 45°C\,. $$

Therefore, the correct answer is Option C: $$45°C$$.