Obtain the mathematical expression for the heat generated in a metallic condactor by electric current (Joule's law).

Explanation

If V is the potential difference applied across a metallic conductor of resistance R, the current through the conductor, given by Ohm's law, is `I=V//R" "...(1)`.The charge passing through the conductor in time t when the current I flows in the conductor is `Q="It " ...(2)`.
The work done in the process is `W=VQ" "...Q`
From Eq. (1) ,(2) and (3), we have,
`W=(IR) (It)=I^2 RT=VIt`
`=V((V)/(R))t=(V^(2))/(R)t`
This work is converted into heat. When I is expressed in ampere, R in ohm, t in second and Vin volt, W is expressed in joule. In that case,
`W=I^(2)RT =VIt=(V^(2))/(R)t` (in joule)
Usually heat energy (H) is expressed in calorie . Using the relation 4.18 J = 1cal , we have
`H=W=(I^(2)RT)/(4.18) ` (in cal)
`=(VIt)/(4.18)` (in cal) `=(V^(2)t)/(4.18R)` (in cal)
This is the required expression.