The heat produced in a wire carrying an electric current depends on the current, the resistance and the time. Assuming that the dependuance is of the product of powers type, guress an eqn. between these quantites uning dimesional analysis. The dimensional formula of resistance is `ML^2 A^(-2) T^(-3)` and heat is a form of energy.

Explanation

Let `H = I^a R^b t^c….(i)`
`[ML^2 T^(-2)] = A^a[ML^2 T^(-3) A^(-2)]^b [T]^c`
`=M^b L^2b T^(-3b+c) A ^(-2b+a)`
Applying the principle of homogeneity of
dimensions, we get
`b =1 , 2b = 2 , -3b +c =-2 ,`
`c = - 2 +3B =- 2+3 =1, and `
`-2b +a =0, a =2b =2xx1`
Putting these values in (i) we get `H = I^2 R^1 t^1`