Heat produced in a current carrying conducting wire depends on current I, resistance R of the wire and time t for which current is passed. Using these facts, obtain the formula for heat energy

Explanation

Suppose, heat energy `H prop I^(a) R^(b) t^(c)`
`:. H= kU^(A)R^(b)t^(c)....(i)`
(where ik is a dimensionless constant).
Now, writing the dimensional formula on both sides,
`M^(1)L^(2)T^(-2)=(A)^(a)(M^(1)L^(2)T^(-3)A^(-2))(T^(c))`
`:.M^(-1)L^(2)T^(-2)=(A)^(a-2b)M^(b)L^(2b)T^(c-3b)`
`[H]M^(1)L^(2)T^(-2)`
`[I]=A[t]=T`
`[R]=M^(1)L^(2)T^(-3)A^(-2)`
Equating the indices on both sides
`a-2b=0,b=1 :. a-2(1)=0 :. a=2`
`-3b+c=2`
`:.c=-2+3=1`
Thus, a = 2, b = 1 and c= 1.
Substituting these values in equation (i), we get
`H=kI^(2)Rt`
`H=kI^(2)Rt` But `k=1 :. H=I(2)Rt`