A material of resistivity ρis formed into a solid, truncated cone of height h and radii r1 and r

step 1 On this problem on current and resistance, we are told that a certain material with resistivity

A material of resistivity ρis formed into a solid, truncated...

A material of resistivity $\rho$ is formed into a solid, truncated cone of height $h$ and radii $r_{1}$ and $r_{2}$ at either end (Fig. $P 25.65 )$ . (a) Calculate the resistance of the cone between the two flat end faces. (Hint: Imagine slicing the cone into very many thin disks, and calculate the resistance of one such disk.) (b) Show that your result agrees with Eq. (25.10) when $r_{1}=r_{2}$ .

Key Concepts

Electrical Resistivity

Electrical resistivity is an intrinsic property of a material that quantifies how strongly it opposes the flow of electric current. In problems involving resistance, resistivity (usually denoted by ?) is used together with the material's geometry to determine the overall resistance encountered by the current passing through the object.

Resistance of a Non-uniform Conductor

When dealing with conductors whose cross-sectional area changes along the length—such as in a truncated cone—the resistance cannot be computed using the simple formula R = ?L/A since the area A varies with position. Instead, the conductor must be analyzed as a continuum of infinitesimally thin slices, each contributing a differential amount of resistance, which are then summed to obtain the total resistance.

Integration Using Differential Elements

Integration using differential elements involves dividing a physical object into infinitesimally small sections, analyzing each section individually, and then summing their contributions to yield a total property, such as resistance. In the context of the truncated cone, the differential resistance of a thin disk is calculated and integrated along the length of the cone to account for the variation in cross-sectional area.

Special Case Consistency Check

A consistency check is performed by verifying that the result for a complex non-uniform geometry reduces to a well-known formula when applied to a special case—in this instance, confirming that the resistance formula for the truncated cone agrees with that for a cylinder when the radii of both ends are equal. This process validates the derived expression by demonstrating its compatibility with established results in simpler, uniform cases.

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