Which of the following is not a dimensionless quantity ?
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Remember that any quantity defined as a ratio of two other quantities with the same units (like relative density, relative permeability, strain, etc.) will be dimensionless. Also, arguments of trigonometric, logarithmic, and exponential functions must be dimensionless.
Step 1: Understanding the Question:
The question asks us to identify which of the given physical quantities has dimensions. A dimensionless quantity is a pure number without any physical units. Step 2: Key Formula or Approach:
We need to analyze the definition and formula for each quantity to determine if it has units and dimensions. Step 3: Detailed Explanation:
(A) Quality factor (Q factor): In the context of resonance, the Q factor is defined as \(Q = 2\pi \times \frac{\text{Maximum energy stored}}{\text{Energy dissipated per cycle}}\). Since it is a ratio of two energy values, their units cancel out, making the Q factor a dimensionless quantity.
(B) Power factor: The power factor in an AC circuit is defined as the cosine of the phase angle (\(\phi\)) between the voltage and current, i.e., \(\cos(\phi)\). The output of any trigonometric function is a pure number, so the power factor is dimensionless.
(C) Relative magnetic permeability (\(\mu_r\)): This is defined as the ratio of the permeability of a medium (\(\mu\)) to the permeability of free space (\(\mu_0\)), i.e., \(\mu_r = \frac{\mu}{\mu_0}\). Since it's a ratio of two quantities with the same units, \(\mu_r\) is dimensionless.
(D) Permeability of free space (\(\mu_0\)): This is a physical constant that relates magnetic fields to electric currents. Its value is \(4\pi \times 10^{-7}\) H/m (henries per meter) or T·m/A (tesla-meters per ampere). Since it has units, it is not a dimensionless quantity. Its dimension is \([M L T^{-2} I^{-2}]\). Step 4: Final Answer:
Permeability of free space (\(\mu_0\)) is the only quantity in the list that has dimensions and units. Therefore, it is not a dimensionless quantity.
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