In each of the following questions, Match column-I and...

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) $\mathrm{A} \rightarrow \mathrm{Q} ; \mathrm{B} \rightarrow \mathrm{P} ; \mathrm{C} \rightarrow \mathrm{R} ; \mathrm{D} \rightarrow \mathrm{S}$

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) $\mathrm{A} \rightarrow \mathrm{Q} ; \mathrm{B} \rightarrow \mathrm{P} ; \mathrm{C} \rightarrow \mathrm{R} ; \mathrm{D} \rightarrow \mathrm{S}$

Key Concepts

Magnetic Field Due to a Circular Arc

A circular arc carrying a current generates a magnetic field at a point (often the center of the arc) that is calculated by integrating the contributions of the current along the arc. This concept is similar to that of a full circular loop but factors in that the current only spans a portion of a complete circle, leading to a proportionate reduction in the magnetic field strength.

Magnetic Field on the Axis of a Current Loop or Coil

The magnetic field along the axis of a circular loop or coil, especially at a point away from the center, is calculated by integrating the contributions from the current elements over the circular path. This concept extends the analysis from the center to other axial points, yielding expressions that include the distance from the center along with the current and radius.

Magnetic Field of a Straight Current-Carrying Conductor

For an infinitely long straight conductor, the magnetic field at a perpendicular distance is derived using the Biot?Savart law. The field has an inverse relationship with distance, and its direction follows the right-hand rule. This concept is central in understanding how linear current distributions generate circular magnetic field lines.

Biot-Savart Law

The Biot?Savart law is a fundamental principle used to calculate the magnetic field produced at a point in space by a small segment of current. It forms the basis for deriving the magnetic field expressions for different current-carrying geometries by integrating the contributions of each current element, taking into account their relative positions and orientations.

Magnetic Field at the Center of a Circular Loop

When current flows through a circular loop, the contributions from the current elements add up symmetrically at the center of the loop. By applying the Biot?Savart law and integrating over the loop, one obtains an expression for the magnetic field at the center, which depends on the current and radius of the loop.

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