Question

Consider a block of copper that is a rectangular prism (a box) with sides 15 cm by 20 cm by 50 cm. The resistivity of copper is 1.68 * 10^-8 ohmM.
What is the ratio of the largest resistance between parallel sides, to the smallest resistance between parallel sides?

Answer 1

The resistance of a conductor can be calculated using the formula:

R = (ρ * L) / A

Where R is the resistance, ρ is the resistivity, L is the length of the conductor, and A is the cross-sectional area of the conductor.

In this case, we need to calculate the resistance between parallel sides of different lengths. Let's call the three sides: L1, L2, and L3.

The lengths of the sides are:
L1 = 15 cm
L2 = 20 cm
L3 = 50 cm

The cross-sectional areas of the sides are:
A1 = (20 cm) * (50 cm) = 1000 cm^2
A2 = (15 cm) * (50 cm) = 750 cm^2
A3 = (15 cm) * (20 cm) = 300 cm^2

Now, we can calculate the resistances:

R1 = (ρ * L1) / A1
R2 = (ρ * L2) / A2
R3 = (ρ * L3) / A3

R1 = (1.68 * 10^-8 ohmM * 15 cm) / 1000 cm^2
R1 = 2.52 * 10^-10 ohms

R2 = (1.68 * 10^-8 ohmM * 20 cm) / 750 cm^2
R2 = 4.48 * 10^-10 ohms

R3 = (1.68 * 10^-8 ohmM * 50 cm) / 300 cm^2
R3 = 2.8 * 10^-8 ohms

The largest resistance is R3, and the smallest resistance is R1. The ratio between them is:

R3 / R1 = (2.8 * 10^-8 ohms) / (2.52 * 10^-10 ohms)
R3 / R1 = 111.11

Therefore, the ratio of the largest resistance between parallel sides to the smallest resistance between parallel sides is approximately 111.