Question

A cylindrical wire of radius $0.5mm$ and conductivity $5\times {10}^{7}S/m$ is subjected to an electric field of $10mV/m$. The expected value of current in the wire will be $x^3\pi mA$. The value of $x$ is ____.

Answer 1

Step 1: Given Data:

Radius of wire $R=0.5mm$

$=5\times {10}^{-4}m$

Conductivity $\sigma =5\times {10}^{7}S/m$

Electric field $E=10mV/m$

$=10\times {10}^{-3}V/m$

Expected value of current $=x^3\pi mA$

Step 2: Calculating the current density:

Current density $J$ is given as,

$J=\sigma E$

Where, $\sigma =$ Conductivity of material

$E=$Electric field

$\therefore J=5\times {10}^7\times 10\times {10}^{-3}$

$=5\times {10}^{5}A/m^2$

Step 3: Calculating the expected current:

Current density $J=\frac{current}{areaofcrosssection}$

$current=J\times {\mathrm{\pi R}}^2$

$=5\times {10}^5\times \pi \times {\left(5\times {10}^{-4}\right)}^2$

$=125\mathrm{\pi }\times {10}^{-3}\mathrm{A}$

$=5^3\mathrm{\pi }\times {10}^{-3}\mathrm{A}$

$=5^3\mathrm{\pi }\mathrm{mA}$

Now, comparing the value of expected current value of $x$is found to be $5$.

Hence, value of $x$is $5$.