Question

In the formula, $X=3Y{Z}^2,X$ and $Z$ have dimensions of capacitance and magnetic field respectively. What are the dimensions of $\mathrm{Y}$ in the MKSQ system?

• (A) $\left[{\mathrm{M}}^{-3}{\mathrm{L}}^{-1}{\mathrm{T}}^3{\mathrm{Q}}^4\right]$
• (B) $[{\mathrm{M}}^{-3}{\mathrm{L}}^{-2}{\mathrm{T}}^4{\mathrm{Q}}^4$
• (C) $\left[{\mathrm{M}}^{-2}{L}^{-2}{\mathrm{T}}^4{\mathrm{Q}}^4\right]$
• (D) $\left[{\mathrm{M}}^{-3}{\mathrm{L}}^{-2}{\mathrm{T}}^4\mathrm{Q}\right]$

Answer 1

The Correct Option is B

Explanation:
The magnetic force (F) experienced by a moving charged particle(q) in a uniform magnetic field (B) is given by$F=qvB\sin \theta$$\Rightarrow B=\frac{F}{qv\sin \theta }$where, $\theta$ is angle between the magnetic field and the velocity of the particle.The dimensions of charge, force and velocity are$[q]=[Q]$$[F]=\left[ML{T}^{-2}\right]$$[\mathrm{v}]=\left[{\mathrm{LT}}^{-1}\right]$$\sin \theta$ is a dimensionless quantity. Now, the dimension of magnetic field is$[B]=\frac{[F]}{|q|v\mid \sin \theta ]}=\frac{\left[ML{T}^{-2}\right]}{[Q]\left[L{T}^{-1}\right]}=\left[M{T}^{-1}{Q}^{-1}\right]$The energy stored in a capacitor is given by$E=\frac{q^2}{2C}$where, $C$ is capacitance.The dimension of energy is$[E]=\left[M{L}^2{T}^{-2}\right]$Thus, the dimension of capacitance is$[C]=\frac{\left[q^2\right]}{[E]}$$\Rightarrow [C]=\frac{\left[Q^2\right]}{\left[M{L}^2{T}^{-2}\right]}$$\Rightarrow [C]=[M^{-1}{L}^{-2}{T}^2{Q}^2]$Given: $\mathrm{X}=3{\mathrm{YZ}}^2$ where, $\mathrm{X}$ and $\mathrm{Z}$ have the dimensions of capacitance and magnetic field respectively. Thus, $[X]=[C]=[M^{-1}{L}^{-2}{T}^2{Q}^2$$[z]=[B]=\left[M{T}^{-1}{Q}^{-1}\right]$Therefore, the dimension of $Y$ is $[Y]=\frac{[X]}{[Z{]}^2}$$\Rightarrow [Y]=\frac{\left[{\mathrm{M}}^{-1}{\mathrm{L}}^{-2}{\mathrm{T}}^2{Q}^2\right]}{{\left[{\mathrm{MT}}^{-1}{\mathrm{Q}}^{-1}\right]}^2}$$\Rightarrow [Y]=\frac{\left[M^{-1}{L}^{-2}{T}^2{Q}^2\right]}{\left|M^2{T}^{-2}{Q}^{-2}\right|}$$\Rightarrow [Y]=\left[M^{-3}{L}^{-2}{T}^4{Q}^4\right]$Hence, the correct option is (B).