The heat produced (H)in a wire carrying an electric currents depend on the current the resistance and the time assuming that the dependence is of the product of powers type guess an equation between these quantities using dimensional analysis heat is a form of energy




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Heat energy produced = power * time  duration = I² R t = V² t / R

H =  I^k  R^m  T^n

Let us try to find the dimensions of  current in known units.
Force = charge * electric field = current * time * voltage /distance
current = Force * distance / (time * voltage)
           = force * distance/(time  * current * resistance)
   I = M L T⁻² * L / (T * I * R)  = M L² T⁻³ I⁻¹ R⁻¹
   I² = M L² T⁻³ R⁻¹
   I = M^1/2  L  T^-3/2  R^-1/2

M L² T⁻²  R^0 = energy =  I^k R^m T^n
            = M^k/2  L^k  T^-3k/2  R^-k/2  *  R^m  * T^n
Equating the powers of the units :
       k = 2        n = 1        m = 1

Hence,  H = I^2 R T

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