Question

the rectangular block below has front - face dimensions of 10 cm by 4 cm, with a depth of 3 cm (figure 1). you will be asked to treat this object as an electrical resistor. rank the block based on its electrical resistance along the three illustrated coordinate directions (x, y, and z). rank from largest to smallest. to rank items as equivalent, overlap them.

Explanation:

Step1: Recall resistance formula

The resistance formula is $R =
ho\frac{l}{A}$, where $
ho$ is the resistivity (constant for a given material), $l$ is the length of the current - path, and $A$ is the cross - sectional area perpendicular to the current flow.

Step2: Calculate resistance for x - direction

For the x - direction, $l_x=10\ cm$ and $A_x = 4\times3\ cm^2 = 12\ cm^2$. So $R_x=
ho\frac{10}{12}=\frac{5}{6}
ho$.

Step3: Calculate resistance for y - direction

For the y - direction, $l_y = 4\ cm$ and $A_y=10\times3\ cm^2 = 30\ cm^2$. So $R_y=
ho\frac{4}{30}=\frac{2}{15}
ho$.

Step4: Calculate resistance for z - direction

For the z - direction, $l_z = 3\ cm$ and $A_z=10\times4\ cm^2 = 40\ cm^2$. So $R_z=
ho\frac{3}{40}$.

Step5: Compare the resistances

To compare $\frac{5}{6}
ho$, $\frac{2}{15}
ho$ and $\frac{3}{40}
ho$, we find a common denominator. The common denominator of 6, 15 and 40 is 120. $\frac{5}{6}
ho=\frac{100}{120}
ho$, $\frac{2}{15}
ho=\frac{16}{120}
ho$, $\frac{3}{40}
ho=\frac{9}{120}
ho$.

Answer:

$x>y>z$