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1.Units, Dimensions and Measurement
hard
Match List $I$ with List $II$
| List $I$ | List $II$ |
| $A$ Torque | $I$ ${\left[\mathrm{M}^1 \mathrm{~L}^1 \mathrm{~T}^{-2} \mathrm{~A}^{-2}\right]}$ |
| $B$ Magnetic fileld | $II$ $\left[\mathrm{L}^2 \mathrm{~A}^1\right]$ |
| $C$ Magnetic moment | $III$ ${\left[\mathrm{M}^1 \mathrm{~T}^{-2} \mathrm{~A}^{-1}\right]}$ |
| $D$ Permeability of free space | $IV$ $\left[\mathrm{M}^1 \mathrm{~L}^2 \mathrm{~T}^{-2}\right]$ |
A$A-I, B-III, C-II, D-IV$
B$A-IV, B-III, C-II, D-I$
C $A-III, B-I, C-II, D-IV$
D$A-IV, B-II, C-III, D-I$
(JEE MAIN-2024)
Solution
${[\tilde{\tau}]=[\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}]=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]}$
${[\mathrm{F}]=[\mathrm{qVB}]}$
$\Rightarrow \mathrm{B}=\left(\frac{\mathrm{F}}{\mathrm{qV}}\right)=\left[\frac{\mathrm{MLT}^{-2}}{\mathrm{ATLT}^{-1}}\right]=\left[\mathrm{MA}^{-1} \mathrm{~T}^{-2}\right]$
${[\mathrm{M}]=[\mathrm{I} \times \mathrm{A}]=\left[\mathrm{AL}^2\right]}$
$\mathrm{B}=\frac{\mu_0}{4 \pi} \frac{\mathrm{Idl} \sin \theta}{\mathrm{r}^2}$
$\Rightarrow[\mu]=\left[\frac{\mathrm{Br}^2}{\mathrm{Idl}}\right]=\left[\frac{\mathrm{MT}^{-2} \mathrm{~A}^{-1} \times \mathrm{L}^2}{\mathrm{AL}}\right]$
$=\left[\mathrm{MLT}^{-2} \mathrm{~A}^{-2}\right]$
${[\mathrm{F}]=[\mathrm{qVB}]}$
$\Rightarrow \mathrm{B}=\left(\frac{\mathrm{F}}{\mathrm{qV}}\right)=\left[\frac{\mathrm{MLT}^{-2}}{\mathrm{ATLT}^{-1}}\right]=\left[\mathrm{MA}^{-1} \mathrm{~T}^{-2}\right]$
${[\mathrm{M}]=[\mathrm{I} \times \mathrm{A}]=\left[\mathrm{AL}^2\right]}$
$\mathrm{B}=\frac{\mu_0}{4 \pi} \frac{\mathrm{Idl} \sin \theta}{\mathrm{r}^2}$
$\Rightarrow[\mu]=\left[\frac{\mathrm{Br}^2}{\mathrm{Idl}}\right]=\left[\frac{\mathrm{MT}^{-2} \mathrm{~A}^{-1} \times \mathrm{L}^2}{\mathrm{AL}}\right]$
$=\left[\mathrm{MLT}^{-2} \mathrm{~A}^{-2}\right]$
Standard 11
Physics
