Mastering Physics Solutions: Dependence of Resistance on Resistor Dimensions Ranking Task

Dependence of Resistance on Resistor Dimensions Ranking Task

Part A = y > z > x
Part B = The resistance is halved.

Solution Below:

The rectangular block below has front-face dimensions of 10 cm by 4 cm, with a depth of 3 cm. You will be asked to treat this object as an electrical resistor.

Mastering Physics Solutions: Dependence of Resistance on Resistor Dimensions Ranking Task

Part A

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.

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):

Rx is in proportion to 3 / 40 = 0.075
Ry is in proportion to 10 / 12 = 0.83
Rz 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

if all of the dimensions of the block double (to become 20 cm wide, 8 cm}tall, and 6 cm deep), what happens to the resistance along each axis?

  • 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.

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