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Units and Measurements question

2023 · 8 Apr · Shift 1 · Q52
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Units and Measurements question

2023 · 8 Apr · Shift 1 · Q52

JEE MainPhysicsUnits and MeasurementsMCQ+4 / −1
Dimension of 1μ0∈0\frac{1}{\mu_{0} \in_{0}}μ0​∈0​1​ should be equal to
  1. A
    T2/L2\mathrm{T}^{2} / \mathrm{L}^{2}T2/L2
  2. B
    T/L\mathrm{T} / \mathrm{L}T/L
  3. C
    L2/T2\mathrm{L}^{2} / \mathrm{T}^{2}L2/T2
  4. D
    L/T\mathrm{L} / \mathrm{T}L/T
View written solutionFree

Correct answer: C

  1. We need the dimensions of 1μ0ε0.\frac{1}{\mu_0\varepsilon_0}.μ0​ε0​1​.

  2. From electromagnetism, the speed of light is related to permeability and permittivity by c=1μ0ε0.c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}.c=μ0​ε0​​1​. Squaring both sides, c2=1μ0ε0.c^2 = \frac{1}{\mu_0\varepsilon_0}.c2=μ0​ε0​1​.

  3. Now, the dimensions of speed are [c]=LT−1.[c] = LT^{-1}.[c]=LT−1. Therefore, [c2]=L2T−2.[c^2] = L^2T^{-2}.[c2]=L2T−2.

  4. Hence, [1μ0ε0]=L2T−2.\left[\frac{1}{\mu_0\varepsilon_0}\right] = L^2T^{-2}.[μ0​ε0​1​]=L2T−2.

  5. Compare with the options:

  • A: T2/L2T^2/L^2T2/L2
  • B: T/LT/LT/L
  • C: L2/T2L^2/T^2L2/T2
  • D: L/TL/TL/T

So the correct option is: C\boxed{\text{C}}C​

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