Question

derive dimensions of magnetic induction

Answer 1

we have the formula  for a force acting on a current carrying conductor.
  
$\vec{F}= I \vec{L} X \vec{B}$

           B = magnetic induction = magnetic field strength.
 Force =  B I L      ie.,  current = I    length  = L  force = newtons.
                           let dimensions of current be [ A ]
  M L T⁻² =  B  A  L

  B = [ M T⁻² A⁻¹ ]
 =========================
we have the formula for magnetic field strength at a point d distance away from a current carrying conductor  as :

     B = μ₀ I / (2 π d)        this is found by Ampere's circuital law.
        μ₀ =  Newtons/Ampere² 

    B = M L T⁻²  A⁻²   * A  L⁻¹
 
dimensions      = M T⁻² A⁻¹
 units         =  Tesla  or  kg / sec²-Amp        or  Newtons/Amp-meter