Resistance of a Thin Square Conducting Sheet

To find the resistance between the two opposite shaded faces of the thin square sheet:

1. Identify the length of the conductor along the direction of current flow:
   The current flows from one shaded face across the sheet to the opposite shaded face. The distance traversed by the current is the length of the side of the square:
   $\ell = L$

2. Determine the cross-sectional area perpendicular to the current flow:
   The face through which the current enters (and leaves) has a width of $L$ and a thickness of $t$. Therefore, the cross-sectional area is:
   $A = L \cdot t$

3. Calculate the resistance ($R$):
   Using the formula for electrical resistance in terms of resistivity $\rho$, length $\ell$, and cross-sectional area $A$:
   $R = \rho \frac{\ell}{A} = \rho \frac{L}{L \cdot t} = \frac{\rho}{t}$

From this result, we see that the resistance $R = \frac{\rho}{t}$:

• Is inversely proportional to the thickness $t$.
• Is completely independent of the side length $L$.

Thus, the resistance is independent of $L$.

Correct Option: C