How to find the dimension formula for inductance and also the dimension for resistance ?

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Sep 8, 2017

Answer:

Dimensions of L, #MT^(-2)L^2A^(-2)#

Dimensions of R,

#ML^2T^(-3)A^(-2)#

Explanation:

Firstly consider resistance.

It's defining equation is, Ohm's law,

#V = IR#
#implies R = V/I#

Now #V# has units of (electric field)*(distance).

But electric field has units (force)/(charge).

Also, charge has dimensions of (current)(time) and force has dimensions (mass)(length)/(time)^2.

Thus, dimensions of #V# is,

#[V] = (LMLT^(-2))/(AT)#
#implies [V] = ML^2T^(-3)A^(-1)#

Current #I# has dimensions #[I] =A#

Thus, dimensions of resistance,

#[R] = [[V]]/[[I]] = ML^2T^(-3)A^(-2)#

For inductance, the defining equation is,

#phi = LI#

But #phi# has units (magnetic field)*(length)^2

Magnetic field from Lorentz force law has units, (Force)(velocity)^(-1)(charge)^(-1)

Therefore, dimensions of magnetic field,

#[B] = (MLT^(-2))/(LT^(-1)AT)#
#implies [B] = (MLT^(-2))/(LA)#
#implies [B] = MT^(-2)A^(-1)#

Therefore dimensions of magnetic flux,

#[phi] = [B]L^2#
#implies [phi] = MT^(-2)L^2A^(-1)#

Thus finally, dimensions of inductance,

#[L] = [[phi]]/[[I]]#
#implies [L] = MT^(-2)L^2A^(-2)#

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