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Units and Measurements question

2024 · 8 Apr · Shift 2 · Q64
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Units and Measurements question

2024 · 8 Apr · Shift 2 · Q64

JEE MainPhysicsUnits and MeasurementsMCQ+4 / −1
If ϵo\epsilon_{\mathrm{o}}ϵo​ is the permittivity of free space and E\mathrm{E}E is the electric field, then ϵoE2\epsilon_{\mathrm{o}} \mathrm{E}^2ϵo​E2 has the dimensions :
  1. A
    [ML−1 T−2][\mathrm{M} \mathrm{L}^{-1} \mathrm{~T}^{-2}][ML−1 T−2]
  2. B
    [ML2 T−2][\mathrm{M} \mathrm{L}^2 \mathrm{~T}^{-2}][ML2 T−2]
  3. C
    [M0L−2TA][\mathrm{M}^{\mathrm{0}} \mathrm{L}^{-2} \mathrm{TA}][M0L−2TA]
  4. D
    [M−1 L−3 T4 A2][\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^4 \mathrm{~A}^2][M−1 L−3 T4 A2]
View written solutionFree

Correct answer: A

  1. We need the dimensions of ε0E2\varepsilon_0 E^2ε0​E2.

  2. Use the fact that electric field is

E=FqE = \frac{F}{q}E=qF​

where

[F]=[MLT−2],[q]=[AT][F] = [M L T^{-2}], \qquad [q] = [A T][F]=[MLT−2],[q]=[AT]

So,

[E]=[F][q]=[MLT−2][AT]=[MLT−3A−1][E] = \frac{[F]}{[q]} = \frac{[M L T^{-2}]}{[A T]} = [M L T^{-3} A^{-1}][E]=[q][F]​=[AT][MLT−2]​=[MLT−3A−1]

Hence,

[E2]=[M2L2T−6A−2][E^2] = [M^2 L^2 T^{-6} A^{-2}][E2]=[M2L2T−6A−2]
  1. Now use Coulomb's law:
F=14πε0q1q2r2F = \frac{1}{4\pi \varepsilon_0} \frac{q_1 q_2}{r^2}F=4πε0​1​r2q1​q2​​

Therefore,

[ε0]=[q]2[F][r]2[\varepsilon_0] = \frac{[q]^2}{[F][r]^2}[ε0​]=[F][r]2[q]2​

Substituting dimensions,

[ε0]=[A2T2][MLT−2] [L2]=[M−1L−3T4A2][\varepsilon_0] = \frac{[A^2 T^2]}{[M L T^{-2}]\,[L^2]} = [M^{-1} L^{-3} T^4 A^2][ε0​]=[MLT−2][L2][A2T2]​=[M−1L−3T4A2]
  1. Multiply [ε0][\varepsilon_0][ε0​] and [E2][E^2][E2]:
[ε0E2]=[M−1L−3T4A2] [M2L2T−6A−2][\varepsilon_0 E^2] = [M^{-1} L^{-3} T^4 A^2]\,[M^2 L^2 T^{-6} A^{-2}][ε0​E2]=[M−1L−3T4A2][M2L2T−6A−2]

Adding powers of corresponding dimensions,

[ε0E2]=[M1L−1T−2][\varepsilon_0 E^2] = [M^{1} L^{-1} T^{-2}][ε0​E2]=[M1L−1T−2]
  1. So,
[ε0E2]=[ML−1T−2]\boxed{[\varepsilon_0 E^2] = [M L^{-1} T^{-2}]}[ε0​E2]=[ML−1T−2]​

This matches option A.

  1. Option check:
  • A: [ML−1T−2][M L^{-1} T^{-2}][ML−1T−2] ✅
  • B: [ML2T−2][M L^2 T^{-2}][ML2T−2] ❌
  • C: incorrect ❌
  • D: [ε0][\varepsilon_0][ε0​] itself, not ε0E2\varepsilon_0 E^2ε0​E2 ❌
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