Dimensional Formula of Resistance

The dimensional formula of Resistance is 

[ M1  L2  T-3 A-2 ]

where [M], [L], [T], and [A] are the fundamental quantities: Mass, Length, Time, and Ampere.

From the above formula, the dimensions of resistance are (1, 2, -3, -2).


How to calculate the dimensional formula of resistance?

We can use Ohm’s law to calculate the dimensional formula of resistance. the mathematic formula of Ohm’s law is


  Voltage (V) = Current (I) * Resistance (R) 
  
  Resistance (R) = Voltage / Current

Now, let’s perform the breakdown of this formula in terms of fundamental quantities:


  Voltage = Work / Charge

Replacing the value of Work with Force * displacement,


  Voltage = (Force * displacement) / Charge

Replacing the value of Force with mass * acceleration,


  Voltage = (mass * acceleration * displacement) / Charge

Also,

  • acceleration = speed / time
  • speed = distance / time
  • acceleration = distance / time2

  Voltage = (mass * distance / time2 * displacement) / ChargeĀ 

                 = (mass * distance * displacement) / (Charge * time2)

Now, the formula of resistance becomes


  Resistance = (mass * distance * displacement) / (Charge * Current * time2)

Also, the formula of Charge is Current * time. Hence, the formula becomes


  Resistance = (mass * distance * displacement) / (Current * time * Current * time2)

                       = (mass * distance * displacement) / (Current2 * time3)

                       = (mass * distance * displacement * time-3 * Current-2)

Dimensionally, we use

  • [M] for mass
  • [L] to represent both distance and displacement
  • [T] for time
  • and [A] for current

Now, the formula becomes


  Resistance = [M1] * [L2] * [T-3] * [A-2]

Hence, the dimensional equation of Resistance is R = [ M1 L2 T-3 A-2 ].


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