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Electricity and Magnetism

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Moderate

Current Density Within a Conical Conductor

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Suppose that we place 10 V across the conical cylindrical conductor shown in figure 1. The left hand side of the conductor has a diameter of $a = 4 \text{ mm}$, and the right hand side of the conductor has a diameter of $b = 2 \text{ mm}$. The conical section of the conductor is $c = 2 \text{ mm}$ wide, and the resistance of the conductor is $R = 250 \; \Omega$.

Figure 1. “Timothy Black. Created for Albert.io. Copyright 2016. All rights reserved.”

What is the magnitude of the current density $J_d$ in section $3$ at a distance $d = 0.5 \text{ mm}$ from the center of the conductor?

A

$J_d = 3.18 \times 10^3$ A/m$^2$

B

$J_d = 5.09 \times 10^4$ A/m$^2$

C

$J_d = 1.27 \times 10^4$ A/m$^2$

D

$J_d = 5.66 \times 10^3$ A/m$^2$