Question

The magnetic flux has the dimension –

& \text{A) M}{{\text{L}}^{2}}{{A}^{-2}} \\\ & \text{B) M}{{\text{L}}^{2}}{{T}^{-1}}{{A}^{-1}} \\\ & \text{C) M}{{\text{T}}^{-2}}{{A}^{-1}} \\\ & \text{D) M}{{\text{L}}^{2}}{{T}^{-2}}{{A}^{-1}} \\\ \end{aligned}$$

Answer 1

We need to understand the given physical quantity thoroughly to determine the dimensional formula of the quantity. We are asked to find the dimensional formula of the magnetic flux in terms of the basic units of the measurement.

Complete step-by-step solution
We know that the dimensional formula of a quantity has to be found by reducing the physical quantity into terms of basic physical quantities such as the mass (M), length (L), time (T), and current (A).
First of all, we need to find the meaning of the physical quantity in the given problem. The magnetic flux is the measure of the magnetic field through a plane surface of area A. It is mathematically given as –
${{\phi }_{B}}=B.A$
Where B is the magnetic field and A is the area of the plane.
Now, we need to find the formula for the magnetic field. It is defined as the force experienced by a charge moving with a velocity ‘v’. It is the force per unit charge per unit velocity. It is given as –
$B=\dfrac{F}{qv}$
Now, let us use the dimensions of each of the physical quantities to derive the dimensional formula of the magnetic flux.
The force is the product of the mass and acceleration as given by Newton’s second law of motion.
i.e.,
$F=ma$
The acceleration is the rate of change of velocity and velocity is the rate of change of displacement. We can now write the force in its most basic form as –
$F=m\dfrac{s}{{{t}^{2}}}$
The charge is the product of current and the time of flow, which can be given as –
$q=It$
Now, we can write the magnetic flux as –

& \phi =BA \\\ & \Rightarrow \phi =\dfrac{F}{qv}A \\\ & \Rightarrow \phi =\dfrac{m\dfrac{s}{{{t}^{2}}}}{It\dfrac{s}{t}}{{s}^{2}} \\\ & \Rightarrow \phi =\dfrac{m{{s}^{2}}}{I{{t}^{2}}} \\\ \end{aligned}$$ Where, m is the mass, s is the distance term, I is the current and t is the time. Now we can write the dimensional formula of the magnetic flux as – $$[\phi ]=[M{{L}^{2}}{{T}^{-2}}{{A}^{-1}}]$$ This is the required dimensional formula of the magnetic flux. **The correct answer is option D.** **Note:** The dimensional formula of two or more physical quantities can be the same. The dimensional formula gives the idea of how the given quantity is related to other basic quantities. The quantities having the same formula may have a different meaning in physical application.