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 $2\, mm$. The end-to-end potential applied to the wire is $50 \,V$. How much energy (in joule) is converted to thermal energy in $100 \,s$ ?

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

Answer 1

Answer: (B)

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