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Q.

Find the dimensional quantity which is equal to the dimensional ratio of magnetic flux and the resistance.

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a

Induced emf

b

Charge

c

Inductance

d

Current

answer is B.

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Detailed Solution

Concept: The dimension of any physical quantity can be calculated by breaking down the physical quantity into basic physical dimensions such as length, mass and time raised to a rational power.
As the magnetic field travels across the specified area, the magnetic flux is defined as such.
Consequently, the following is the dimensional formula for magnetic flux:

[ϕ] = [B] X [Area]

Now, the magnetic field in terms of magnetic force is given by,

F = q (\vec{v} X \vec{B})

Therefore,

[ϕ] = \frac{[F] [A]}{[q] [v]}

On putting dimensional formula for each term, we get

= \frac{[M^{1} L^{1} T^{- 2}] [L^{2}]}{{[A}^{1} T^{1}] [{LT}^{- 1}]}

= [M^{1} L^{2} T^{- 2} A^{- 1}]

………(1)
Now, according to ohm’s law, resistance is given by,

R = \frac{V}{I}

And, the dimensional formula of resistance is given by,

[R] = \frac{[V]}{[I]}

= \frac{[M^{1} L^{2} A^{- 1} T^{- 3}]}{[A]}

= [M L^{2} T^{- 3} A^{- 2}]

……………..(2)
On dividing equations (1) by (2) , we get
i.e.

\frac{[ϕ]}{[R]} = \frac{[M^{1} L^{2} T^{- 2} A^{- 1}]}{[M^{1} L^{2} T^{- 3} A^{- 2}]} = [AT]

Since the charge is given by,

q = It

(Here ‘I’ represents the current)
This implies that the dimensional formula of charge is

[AT]

Hence, the correct option is 2.

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