Question

The dimensions [MLT^-2A^-2] belong to the:

• A. Magnetic flux
• B. Self Inductance
• C. Magnetic permeability
• D. Electric permittivity

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Dimensional analysis is a tool used to check the relationship between various physical quantities by expressing them in terms of fundamental dimensions like Mass (\(M\)), Length (\(L\)), Time (\(T\)), and Current (\(A\)).
Magnetic permeability (\(\mu_0\)) is a measure of the ability of a material to support the formation of a magnetic field within itself.
Key Formula or Approach:
A reliable way to find the dimensions of \(\mu_0\) is using the formula for the force (\(F\)) between two parallel long wires carrying currents \(I_1\) and \(I_2\) separated by a distance \(r\):
\[ \frac{F}{l} = \frac{\mu_0 I_1 I_2}{2\pi r} \]
where \(l\) is the length of the wires.
Step 2: Detailed Explanation:
Isolating \(\mu_0\) from the formula:
\[ \mu_0 = \frac{2\pi r F}{l I_1 I_2} \]
Now, let's substitute the dimensions for each physical quantity:
1. Force (\(F\)): \([MLT^{-2}]\)
2. Distance (\(r\)) and length (\(l\)): \([L]\)
3. Current (\(I_1, I_2\)): \([A]\)
4. \(2\pi\) is a dimensionless constant.
Substituting these dimensions:
\[ [\mu_0] = \frac{[L] [MLT^{-2}]}{[L] [A] [A]} \]
The length dimension \([L]\) in the numerator and denominator cancels out:
\[ [\mu_0] = \frac{[MLT^{-2}]}{[A^2]} \]
\[ [\mu_0] = [MLT^{-2}A^{-2}] \]
This matches the dimensions provided in the question.
Step 3: Final Answer:
The given dimensions \([MLT^{-2}A^{-2}]\) belong to magnetic permeability.