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Question 45

In an electromagnetic system, a quantity defined as the ratio of electric dipole moment and magnetic dipole moment has dimension of $$[M^P L^Q T^R A^S]$$. The value of P and Q are :

For any physical quantity we can express its dimension in the form $$[M^{P} L^{Q} T^{R} A^{S}]$$, where $$M$$ stands for mass, $$L$$ for length, $$T$$ for time and $$A$$ for electric current.

Electric dipole moment
Definition : $$p_{e}=q\,d$$, where $$q$$ is charge and $$d$$ is the separation between the charges.
Dimension of charge : $$[q]=[A\,T]$$ (current × time).
Therefore
$$[p_{e}] = [A\,T]\,[L] = [M^{0} L^{1} T^{1} A^{1}]$$

Magnetic dipole moment
For a current loop, $$m_{m}=I\,A_{\text{loop}}$$, where $$I$$ is current and $$A_{\text{loop}}$$ is the area of the loop.
Area has dimension $$[L^{2}]$$, hence
$$[m_{m}] = [A]\,[L^{2}] = [M^{0} L^{2} T^{0} A^{1}]$$

Required ratio
$$\frac{p_{e}}{m_{m}}$$ has dimension $$\frac{[M^{0} L^{1} T^{1} A^{1}]}{[M^{0} L^{2} T^{0} A^{1}]}$$ = $$[M^{0} L^{1-2} T^{1-0} A^{1-1}]$$ = $$[M^{0} L^{-1} T^{1} A^{0}]$$.

Comparing with $$[M^{P} L^{Q} T^{R} A^{S}]$$:
$$P=0,\; Q=-1,\; R=1,\; S=0$$.

Hence the values of $$P$$ and $$Q$$ are 0 and −1 respectively, which corresponds to Option D.

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