(II) A rectangular solid made of carbon has sides of lengths 1.0 cm, 2.0 cm, and 4.0 cm, lying a

(II) A rectangular solid made of carbon has sides of lengths...

(II) A rectangular solid made of carbon has sides of lengths $1.0 \mathrm{cm}, 2.0 \mathrm{cm},$ and $4.0 \mathrm{cm},$ lying along the $x, y,$ and $z$ axes, respectively (Fig. $35 ) .$ Determine the resistance for current that passes through the solid in $(a)$ the $x$ direction, $(b)$ the $y$ direction, and $(c)$ the $z$ direction. Assume the resistivity is $\rho=3.0 \times 10^{-5} \Omega \cdot \mathrm{m}$

Key Concepts

Directional Dependence in Anisotropic Conduction

In cases where a solid has different dimensions along different axes, the resistance can vary depending on the direction of current flow. This directional dependence reflects how the effective length and cross-sectional area change with orientation, making it essential to analyze each case individually when calculating resistance.

Impact of Geometry

The geometry of a conductor, particularly the length of the path that current takes and the cross-sectional area through which it flows, plays a critical role in determining its resistance. Changes in the dimensions that affect these parameters will alter the resistance according to the relationship R = ?L/A.

Electrical Resistance

Electrical resistance is the measure of how a material opposes the flow of electric current. It is determined by both the material properties and its physical dimensions, typically calculated using the formula R = ?L/A, where ? is the resistivity, L is the length of the current path, and A is the cross-sectional area perpendicular to the current.

Resistivity

Resistivity is an intrinsic property of a material that quantifies how strongly it resists current flow. It is independent of the shape or size of the material and is usually measured in ohm-meters (?·m). Resistivity is used to predict the resistance of objects when combined with their geometric dimensions.

Transcript

00:01 Hi, alex b here with another question on electric currents and resistance from chapter 25 of geoencolle's physics for scientists and engineers.

00:10 So the problem tells us that we have a chunk of carbon, which i've already drawn here, which measures 1 centimeter by 2 centimeters by 4 centimeters.

00:19 And the question asks us to calculate the resistance for current passing along various directions through this chunk.

00:26 So the only additional piece of information that we need is given to us in the question, and it tells us, the resistivity of this carbon is going to be assumed to be 3 .0 times 10 to the minus 5 oms times meters.

00:41 We're going to plug this into our equation for resistance in terms of resistivity and physical dimensions where resistance is equal to resistivity times the length divided by the area.

00:55 But to know what the length is and what the area is, we need to know which direction we're going through.

01:01 So the first part of the problem asks us to calculate this going through the x direction.

01:06 In this case, resistivity is the same as in every direction, 3 .0 times 10 to the minus 5.

01:15 But the length here is the x -dimensional length, which in this case is 1 .0 times 10 to the minus 2 meters.

01:23 Here i'm going to use meters everywhere, that way we get the right answer at the end in the right units.

01:28 In this case, now the area is going to be the dimension times the z dimension...

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