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

step 1 Hi everyone here it is given truncated coal to find its resistance we have to consider an elemen

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. $\mathbf{P 2 5 . 5 9}$ ). (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 $r_{1}=r_{2}$

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. $\mathbf{P 2 5 . 5 9}$ ). (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

Resistivity

Resistivity is a fundamental material property that quantifies how strongly a material opposes the flow of electric current. It is a proportionality factor in the relationship R = ?L/A, where R is the resistance, ? is the resistivity, L is the length, and A is the cross-sectional area. In systems with variable geometries, like a truncated cone, ? remains constant while the area changes, necessitating integration over the length.

Resistance in Non?Uniform Cross-Sections

When dealing with objects whose cross-sectional areas vary along their length, calculating resistance involves integrating the differential resistance elements. Each thin slice or disk of thickness dx has a resistance dR = ? dx/A(x), and summing these elements over the whole length yields the total resistance. This approach is essential when the usual formula R = ?L/A does not directly apply because of the changing area.

Truncated Cone Geometry

A truncated cone is a three-dimensional shape with circular faces of different radii, where the radius changes linearly along the height of the cone. This linear variation is crucial for setting up the relation between the position along the cone and its local radius, which in turn determines the area A(x) used in the integration of the resistance.

Integration Techniques in Electromagnetism

Integration techniques are used to sum contributions from infinitesimally small elements in problems where physical quantities vary continuously with position. In the context of resistance calculations, the method of slicing the material into thin disks and integrating dR = ? dx/A(x) allows for the systematic treatment of non-uniform cross-sectional areas. This technique ensures that the overall resistance calculation accurately reflects the geometry of the object.

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