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.

Key concepts

Straight Conductor

Understanding the magnetic field induction created by a straight conductor is a fundamental aspect of electromagnetism. When a straight wire carries an electric current, a magnetic field is generated around the wire. This field forms concentric circles perpendicular to the wire. The intensity of the magnetic field decreases with distance from the wire. Mathematically, the magnetic field induction due to a straight conductor at a perpendicular distance \( r \) from the wire can be expressed as:

• \( B = \frac{\mu_0 I}{2 \pi r} \)

In this formula, \( \mu_0 \) is the permeability of free space, \( I \) is the current flowing through the conductor, and \( r \) is the perpendicular distance from the conductor to the point where the field is measured.
This formula helps us predict how strong the magnetic field will be at any given point surrounding the conductor. The closer you are to the wire, the stronger the magnetic field will be. Understanding this phenomenon is crucial for applications like electromagnets and magnetic sensors.

Current Carrying Loop

The magnetic field induction at the center of a current-carrying loop is another key concept. When current flows through a circular loop, it generates a magnetic field within and outside the loop. The magnetic field at the loop's center is particularly significant since it tends to be quite strong.
The expression for the magnetic field induction at the center of a loop of radius \( r \) is given by:

• \( B = \frac{\mu_0 I}{2r} \)

Here, \( \mu_0 \) represents the permeability of free space, \( I \) is the current in the loop, and \( r \) is the radius of the loop.
This formula shows that the magnetic field strength is directly proportional to the current and inversely proportional to the loop radius. Larger currents generate stronger magnetic fields, while larger loops result in weaker fields at their centers. This principle is applied in devices like solenoids and inductors, which utilize loops of wire to create strong magnetic fields.

Current Carrying Coil

When dealing with a coil made of multiple loops, the magnetic field characteristics change slightly. A coil is essentially a stack of circular loops, known as turns, that enhance the magnetic effect. For a current-carrying coil placed along an axis, the magnetic field can be measured at various points along this axis.
If you are considering a distance \( r \) from the center of a coil, the expression for the magnetic field induction changes to:

• \( B = \frac{\mu_0 I}{4\sqrt{2}\pi r} \)

The factor \( \sqrt{2} \) introduces additional complexity that comes with the geometry of coils, significantly affecting the strength of the magnetic field depending on the positioning along the axis.
Coils are widely used in electromagnets and transformers to produce strong, controlled magnetic fields. The unique property of these fields means they are pivotal in many electrical applications, from motors to inductors.

Circular Arc

A less common, yet equally fascinating scenario is the magnetic field induction due to a circular arc. When a current flows through a section of a circle, or an arc, the magnetic field generated behaves differently. At the center of the circle from which the arc is a part, the field's strength is impacted by both the length of the arc and the radius.
The formula for the magnetic field at the center of a circular arc is defined by:

• \( B = \frac{\mu_0 I}{4 \pi r} \)

This equation tells us that the magnetic field's intensity is not only influenced by the current and radius but also by the segment of the circle composed by the arc.
An arc's magnetic influence is more often analyzed in physics to understand complex field distributions. For practical purposes, this principle helps in designing curved solenoids and bent wire systems. Understanding the nuances of such configurations allows for innovative solutions in electronic and electromagnetic setups.