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Electricity and Magnetism

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Current Density Ratio Within a Conical Conductor

EANDM-FEOMVX

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 ratio $R = \cfrac{J_3}{J_1}$ of the current density a distance $d = 0.5 \text{ mm}$ from the center of the conductor in region $3$ to the current density a distance $d$ from the center of the conductor in region $1$?

A

$R = 1$

B

$R = 2$

C

$R = 3$

D

$R = 4$