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QJEEM23Phf3b1: JEE Main 2023 Physics - Class 11 JEEM23Phf3b1 Solution: If the velocity of light , universal gravitational constant and …

If the velocity of light c\mathrm{c} , universal gravitational constant G\mathrm{G} and Planck's constant h\mathrm{h} are chosen as fundamental quantities. The dimensions of mass in the new system is :

Multiple Choice Options

  • A [h1c1G−1]\left[\mathrm{h}^{1} \mathrm{c}^{1} \mathrm{G}^{-1}\right]
  • B [h−1/2c1/2G1/2]\left[\mathrm{h}^{-1 / 2} \mathrm{c}^{1 / 2} \mathrm{G}^{1 / 2}\right]
  • C [h1/2c1/2G−1/2]\left[\mathrm{h}^{1 / 2} \mathrm{c}^{1 / 2} \mathrm{G}^{-1 / 2}\right]
  • D [h1/2c−1/2G1]\left[\mathrm{h}^{1 / 2} \mathrm{c}^{-1 / 2} \mathrm{G}^{1}\right]

Step-by-Step Verified Solution

amp;[M]=[G]x[h]y[c]zamp;[M]=[M−1 L3 T−2]x[ML2 T−1]y[LT−1]zamp;[M1 L0 T0]=[M−x+y][L3x+2y+z][T−2x−y−z]amp;y−x=1......(1)amp;3x+2y+z=0.......(2)amp;−2x−y−z=0........(3)\begin{aligned}& {[\mathrm{M}]=[\mathrm{G}]^{\mathrm{x}}[\mathrm{h}]^{\mathrm{y}}[\mathrm{c}]^{\mathrm{z}}} \\\\& {[\mathrm{M}]=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]^x\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^y\left[\mathrm{LT}^{-1}\right]^z} \\\\& {\left[\mathrm{M}^1 \mathrm{~L}^0 \mathrm{~T}^0\right]=\left[\mathrm{M}^{-x+y}\right]\left[\mathrm{L}^{3 x+2 y+z}\right]\left[\mathrm{T}^{-2 x-y-z}\right]} \\\\& \mathrm{y}-\mathrm{x}=1 ......(1) \\\\& 3 \mathrm{x}+2 \mathrm{y}+\mathrm{z}=0 .......(2) \\\\& -2 \mathrm{x}- \mathrm{y}-\mathrm{z}=0 ........(3)\end{aligned}

On solving, x=−12,y=12,z=12\mathrm{x}=-\frac{1}{2}, \mathrm{y}=\frac{1}{2}, \mathrm{z}=\frac{1}{2}

So m=hcG\mathrm{m}=\sqrt{\frac{\mathrm{hc}}{\mathrm{G}}}

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