Dependence of Resistance on Resistor Dimensions Ranking Task

Part A = y > z > x

Part B = The resistance is halved.

**Solution Below:**

Part A

Rank from largest to smallest. To rank items as equivalent, overlap them.

Resistance is given by R = ρ * l / A, where ρ is the resistivity, l is the length being passed through, and A is the cross sectional area.

In the x direction, length is 3cm and the cross sectional area is 10cm & 4cm = 40cm. In the y direction, length is 10cm and the cross sectional area is 3cm * 4cm = 12cm. In the z direction, length is 4cm and the cross sectional area is 3cm * 10cm = 30cm. So you have (the material is the same for all directions, so ρ will not affect the ranking):

R_{x} is in proportion to 3 / 40 = 0.075

R_{y} is in proportion to 10 / 12 = 0.83

R_{z} is in proportion to 4 / 30 = 0.13

So the order (from greatest to least resistance) is y > z > x

y > z > x

Part B

- The resistance quadruples.
- The resistance doubles.
- The resistance stays the same.
- The resistance is halved.
- The resistance is quartered.

The resistance will be halved. In the original, the cross sectional area is given by i * j and length by l. The proportional resistance is therefore l/A = l/ij. Doubling the proportions gives you 2l / (2i * 2j) = 2l/4ij = 1/2l/ij = 1/2l/A. So the resistance is halved.

The resistance is halved.