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Which two of the following five physical parameters have the same dimensions ?
(a) Energy density
(b) Refractive index
(c) Dielectric constant
(d) Young's modulus
(e) Magnetic field
(a) Energy density
(b) Refractive index
(c) Dielectric constant
(d) Young's modulus
(e) Magnetic field
A
(a) and (d)
B
(a) and (e)
C
(b) and (d)
D
(c) and (e)
Detailed Solution
Energy density $= \dfrac{\text{energy}}{\text{volume}} = \dfrac{[ML^2T^{-2}]}{[L^3]} = [ML^{-1}T^{-2}]$
Young's modulus $= \dfrac{\text{stress}}{\text{strain}} = \dfrac{\text{force/area}}{\text{dimensionless}} = \dfrac{[MLT^{-2}]}{[L^2]} = [ML^{-1}T^{-2}]$
So energy density and Young's modulus have the same dimensions, $[ML^{-1}T^{-2}]$.
Refractive index is a ratio of two speeds and the dielectric constant is a ratio of two permittivities, so both are dimensionless.
Magnetic field $B = \dfrac{F}{qv}$ has dimensions $[MT^{-2}A^{-1}]$, which matches none of the others.
Among the given pairs, the pair with the same dimensions is (a) energy density and (d) Young's modulus.
Young's modulus $= \dfrac{\text{stress}}{\text{strain}} = \dfrac{\text{force/area}}{\text{dimensionless}} = \dfrac{[MLT^{-2}]}{[L^2]} = [ML^{-1}T^{-2}]$
So energy density and Young's modulus have the same dimensions, $[ML^{-1}T^{-2}]$.
Refractive index is a ratio of two speeds and the dielectric constant is a ratio of two permittivities, so both are dimensionless.
Magnetic field $B = \dfrac{F}{qv}$ has dimensions $[MT^{-2}A^{-1}]$, which matches none of the others.
Among the given pairs, the pair with the same dimensions is (a) energy density and (d) Young's modulus.