If H(heat) depends on I, R, t, then find the formula for H using dimensional analysis

Dear student

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

H =  I^k  R^m  Tⁿ

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 L²T^-3 I^-2 R^-1
   I² = M L² T^-3 R^-1
   I = M^1/2  L  T^-3/2  R^-1/2

M L² T⁻²  R⁰ = energy =  I^k R^m Tⁿ
            = M^k/2  L^k  T^-3k/2  R^-k/2  *  R^m  * Tⁿ
   
Equating the powers of the units :

 k = 2        n = 1        m = 1

Hence,  H = I² R T
Regards