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Chapter 1: Problem 143

The dimensional formula of magnetic flux is ............ (a) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2} \mathrm{~A}^{-1}\) (b) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{1} \mathrm{~A}^{2}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2} \mathrm{~A}^{2}\) (d) \(\mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^{1} \mathrm{~A}^{2}\)

Short Answer

Expert verified
The dimensional formula of magnetic flux is \(\mathrm{M^{1} L^{2} T^{-2} A^{-1}}\).

Step by step solution

01

Determine the dimensions of magnetic field (B)

We know that the magnetic field B has units of Tesla. Tesla is defined as Weber(Φ) / square meter, which means B has the dimensions of magnetic flux per area. To make it easier, let's write the dimensions of magnetic field (B) as \(\mathrm{[B]}\).
02

Determine the dimensions of area (A)

The dimensions of area (A) is given as square meter, which is represented as \(\mathrm{[A] = L^2}\), where L represents length.
03

Determine the dimensions of cos(θ)

Since the cosine function is a dimensionless quantity, the dimensions of cos(θ) are \(\mathrm{[cos(\theta)] = 1}\).
04

Get the dimensional formula of magnetic flux (Φ)

Now that we have the dimensions of B, A, and cos(θ), we can find the dimensions of magnetic flux (Φ): \[\mathrm{[\Phi] = [B] \times [A] \times [cos(\theta)]}\] As the dimension of cos(θ) is 1, it will not affect the dimensions. So, we can ignore it and write \(\mathrm{[\Phi] = [B] \times [A]}\).
05

Substitute the known dimensions

From step 1, we know the dimensions of B: \[\mathrm{[B] = \frac{[\Phi]}{[A]}}\] Now, substitute the dimensions of magnetic flux per area \((\mathrm{M^{1} L^{2} T^{-2} A^{-1}})\) for B and the dimensions of area \((\mathrm{L^2})\) for A: \[\mathrm{[\Phi] = (\frac{M^{1} L^{2} T^{-2} A^{-1}}{L^{2}}) \times L^{2}}\]
06

Solve for the dimensions of magnetic flux (Φ)

When we cancel out the L^2 in the numerator and denominator, we will get the following dimensional formula for magnetic flux: \[\mathrm{[\Phi] = M^{1} L^{2} T^{-2} A^{-1}}\] Therefore, the correct answer is option (a) \(\mathrm{M^{1} L^{2} T^{-2} A^{-1}}\).

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