Question

A circular wire has current density $J=\left(2\times 10^{10}A/m^2\right)r^2$ where $1$ is the radial distance, out of the wire radius is $2mm$. The end-to-end potential applied to the wire is $50V$. How much energy (in joule) is converted to thermal energy in $100s$ ?

• A. $1200\pi$
• B. $800\pi$
• C. $3200\pi$
• D. $600\pi$

Answer 1

Answer: (B)

Heat produced in a section of width $dr$ at a distance $r$ from centre is $H=\underset{0}{\overset{R}{\int }}V\cdot t\cdot j\cdot 2\pi rdr$
Here, $K=50V,t=100s$
$j=2\times 10^{10}{r}^2,R=2\times 10^{-3}m$
So, $H=\underset{0}{\overset{R}{\int }}50\times 100\times 2\times 10^{10}{r}^2\times 2\pi r\times dr$
$=\underset{0}{\overset{R}{\int }}2\pi \times 10^{14}\cdot r^3\cdot dr=2\pi \times 10^{14}\times {\left[\frac{r^4}{4}\right]}_0^R$
$=\frac{\pi }{2}\times 10^{14}\times {\left(2\times 10^{-3}\right)}^4$
$=8\times \pi \times 10^{14}\times 10^{-12}=800\pi J$