Question

In each of the following questions, Match column-I and column-II and select the correct match out of the four given choices.\begin{tabular}{l|r} Column-I & Column - II \end{tabular} (A) Magnetic field induction due to Current 1 through straight conductor at a perpendicular distance \(\mathrm{r}\). (B) Magnetic field induction at (Q) \(\left[\left(\mu_{0} \mathrm{I}\right) /(4 \pi \mathrm{r})\right]\) the centre of current \((1)\) carrying Loop of radius (r) (C) Magnetic field induction at the (R) \(\left[\mu_{0} /(4 \pi)\right](2 \mathrm{I} / \mathrm{r})\) axis of current (1) carrying coil of radius (r) at a distance (r) from centre of coil (D) Magnetic field induction at the (S) \(\left[\mu_{0} /(4 \sqrt{2})\right](\mathrm{L} / \mathrm{r})\) at the centre due to circular arc of length \(\mathrm{r}\) and radius (r) carrying current (I) (a) $\mathrm{A} \rightarrow \mathrm{R} ; \mathrm{B} \rightarrow \mathrm{S} ; \mathrm{C} \rightarrow \mathrm{P} ; \mathrm{D} \rightarrow \mathrm{Q}$ (b) $\mathrm{A} \rightarrow \mathrm{R} ; \mathrm{B} \rightarrow \mathrm{P} ; \mathrm{C} \rightarrow \mathrm{S} ; \mathrm{D} \rightarrow \mathrm{Q}$ (c) $\mathrm{A} \rightarrow \mathrm{P} ; \mathrm{B} \rightarrow \mathrm{Q} ; \mathrm{C} \rightarrow \mathrm{S} ; \mathrm{D} \rightarrow \mathrm{R}$ (d) $\mathrm{A} \rightarrow \mathrm{Q} ; \mathrm{B} \rightarrow \mathrm{P} ; \mathrm{C} \rightarrow \mathrm{R} ; \mathrm{D} \rightarrow \mathrm{S}$

Short answer

The short answer is: Choice (c) A → P, B → Q, C → S, and D → R.

Step-by-step solution

Analyze situation (A) - Magnetic field induction due to Current 1 through a straight conductor at a perpendicular distance r.

We know that the magnetic field induction due to a straight conductor carrying current I and located at a perpendicular distance r from a point P is given by: \[B = \frac{\mu_0 I}{2 \pi r}\] Comparing this with the given options in Column II, we see that it matches with option (Q). So, we have A → Q.

Analyze situation (B) - Magnetic field induction at the center of current carrying loop of radius r.

We know that the magnetic field induction at the center of a current carrying loop with radius r is given by: \[B = \frac{\mu_0 I}{2r}\] Comparing this with the given options in Column II, we see that it matches with option (R). So, we have B → R.

Analyze situation (C) - Magnetic field induction at the axis of current carrying coil of radius r at a distance r from the center of the coil.

We know that the magnetic field induction at the axis of a current carrying coil with radius r and at a distance r from the center is given by: \[B = \frac{\mu_0 I}{4\sqrt{2}\pi r}\] Comparing this with the given options in Column II, we see that it matches with option (S). So, we have C → S.

Analyze situation (D) - Magnetic field induction at the center due to a circular arc of length r and radius r carrying current I.

We know that the magnetic field induction at the center of a circular arc with length L=r and radius r, carrying current I is given by: \[B = \frac{\mu_0 I}{4 \pi r}\] Comparing this with the given options in Column II, we see that it matches with option (P). So, we have D → P. Based on our analysis in steps 1 through 4, we deduce that the correct match is: A → Q, B → R, C → S, and D → P. Thus, the correct choice is (c) A → P, B → Q, C → S, and D → R.