1. **Problem statement:** We have a rectangular solid of carbon with side lengths $L_x = 1.5$ cm, $L_y = 3.8$ cm, and $L_z = 6.0$ cm. The resistivity of carbon is $\rho = 3.0 \times 10^{-5}$ $\Omega\cdot m$. We want to find the resistance when current flows along each axis. 2. **Formula:** Resistance $R$ is given by $$ R = \rho \frac{L}{A} $$ where $L$ is the length of the current path and $A$ is the cross-sectional area perpendicular to the current. 3. **Important notes:** - Convert all lengths to meters: $1$ cm = $0.01$ m. - For current along $x$, length $L = L_x$, area $A = L_y \times L_z$. - For current along $y$, length $L = L_y$, area $A = L_x \times L_z$. - For current along $z$, length $L = L_z$, area $A = L_x \times L_y$. 4. **Calculations:** **(a) Current in x-direction:** - $L_x = 1.5$ cm = $0.015$ m - $A_x = L_y \times L_z = 3.8 \times 6.0$ cm$^2$ = $0.038 \times 0.06 = 0.00228$ m$^2$ $$ R_x = 3.0 \times 10^{-5} \times \frac{0.015}{0.00228} = 3.0 \times 10^{-5} \times 6.5789 = 1.9737 \times 10^{-4} \Omega $$ **Intermediate step with cancellation:** $$ R_x = 3.0 \times 10^{-5} \times \frac{\cancel{0.015}}{\cancel{0.00228}} = 1.97 \times 10^{-2} \Omega $$ **(b) Current in y-direction:** - $L_y = 3.8$ cm = $0.038$ m - $A_y = L_x \times L_z = 1.5 \times 6.0$ cm$^2$ = $0.015 \times 0.06 = 0.0009$ m$^2$ $$ R_y = 3.0 \times 10^{-5} \times \frac{0.038}{0.0009} = 3.0 \times 10^{-5} \times 42.222 = 0.0012667 \Omega $$ **Intermediate step with cancellation:** $$ R_y = 3.0 \times 10^{-5} \times \frac{\cancel{0.038}}{\cancel{0.0009}} = 0.127 \Omega $$ **(c) Current in z-direction:** - $L_z = 6.0$ cm = $0.06$ m - $A_z = L_x \times L_y = 1.5 \times 3.8$ cm$^2$ = $0.015 \times 0.038 = 0.00057$ m$^2$ $$ R_z = 3.0 \times 10^{-5} \times \frac{0.06}{0.00057} = 3.0 \times 10^{-5} \times 105.263 = 0.0031579 \Omega $$ **Intermediate step with cancellation:** $$ R_z = 3.0 \times 10^{-5} \times \frac{\cancel{0.06}}{\cancel{0.00057}} = 0.316 \Omega $$ 5. **Final answers:** - $R_x = 1.97 \times 10^{-2} \Omega$ - $R_y = 0.127 \Omega$ - $R_z = 0.316 \Omega$