Two wires of same length and thickness having specific resistances 6 $$\Omega$$ cm and 3 $$\Omega$$ cm respectively are connected in parallel. The effective resistivity is $$\rho$$ $$\Omega$$ cm. The value of $$\rho$$ to the nearest integer, is ___.

Two wires of the same length $$l$$ and the same thickness (cross-sectional area $$A$$) have specific resistivities $$\rho_1 = 6\,\Omega\,\text{cm}$$ and $$\rho_2 = 3\,\Omega\,\text{cm}$$. Their resistances are $$R_1 = \frac{\rho_1 l}{A}$$ and $$R_2 = \frac{\rho_2 l}{A}$$.

When connected in parallel, the equivalent resistance is $$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{A}{\rho_1 l} + \frac{A}{\rho_2 l} = \frac{A}{l}\left(\frac{1}{6} + \frac{1}{3}\right) = \frac{A}{l} \cdot \frac{1}{2}$$, so $$R_{eq} = \frac{2l}{A}$$.

The parallel combination of two identical-geometry wires is equivalent to a single conductor of length $$l$$ and cross-section $$2A$$. The effective resistivity satisfies $$R_{eq} = \frac{\rho \cdot l}{2A}$$, giving $$\frac{2l}{A} = \frac{\rho l}{2A}$$, hence $$\rho = 4\,\Omega\,\text{cm}$$.

The value of $$\rho$$ is $$4$$.