A) \[\Omega \]
B) \[\Omega \]
C) \[\Omega \]
D) \[1:{{2}^{1/3}}\]
Correct Answer: C
Solution :
According to Biot-Savarts law, the magnetic field due to current element \[idl\] distance r from P is \[B=\frac{{{\mu }_{0}}}{4\pi }\frac{idl\sin \theta }{r}\] For \[\theta ={{90}^{o}},\sin \theta =1\] \[\therefore \] \[B=\frac{{{\mu }_{0}}idl}{4\pi r}\] \[\Rightarrow \] \[{{\mu }_{0}}=\frac{B\times 4\pi {{r}^{2}}}{idl}\] Putting the dimensions, \[B=[M{{T}^{-2}}{{A}^{-1}}]\] \[r=[L]i=[A],dl=[L],\] we have Dimensions of \[{{\mu }_{0}}=\frac{[M{{T}^{-2}}{{A}^{-1}}][{{L}^{2}}]}{[A][L]}\] \[{{\mu }_{0}}=[ML{{T}^{-2}}{{A}^{-2}}]\] Alternative: Magnetic field due to straight solenoid is \[B={{\mu }_{0}}nI\] \[\therefore \] \[{{\mu }_{0}}=\frac{B}{nI}\] \[=\frac{[ML{{T}^{-2}}]/[AL]}{[{{L}^{-1}}A]}\] \[=[ML{{T}^{-2}}{{A}^{-2}}]\] Note: It is worth noting that constants such as 5, \[\pi \] or trigonometrical functions such as \[\sin \theta ,\cos \theta \] etc., have no units and dimensions.
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