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NEET PHYSICSMOVING CHARGES AND MAGNETISMHard

Question

A wire carrying current II has the shape as shown in the adjoining figure. Linear parts of the wire are very long and parallel to the X-axis while the semicircular portion of radius RR is lying in the Y-Z plane. The magnetic field at point OO is:

A

μ0I4πR(πi^+2k^)\frac{\mu_0 I}{4\pi R}(\pi \hat{i} + 2\hat{k})

B

−μ0I4πR(πi^−2k^)-\frac{\mu_0 I}{4\pi R}(\pi \hat{i} - 2\hat{k})

C

−μ0I4πR(πi^+2k^)-\frac{\mu_0 I}{4\pi R}(\pi \hat{i} + 2\hat{k})

D

μ0I4πR(πi^−2k^)\frac{\mu_0 I}{4\pi R}(\pi \hat{i} - 2\hat{k})

Step-by-Step Solution

To find the total magnetic field at point OO, we consider the three segments of the wire independently and use the principle of superposition:

  1. Semi-infinite straight wires: There are two very long (semi-infinite) linear segments parallel to the X-axis. According to the sources, the magnetic field due to a semi-infinite wire at a distance RR from its end is B=μ0I4πRB = \frac{\mu_0 I}{4\pi R} . Applying the right-hand thumb rule for the given geometry, both segments produce a field at OO in the negative Z-direction (−k^-\hat{k}). Total field from linear parts = 2×(μ0I4πR)(−k^)=−2μ0I4πRk^2 \times \left(\frac{\mu_0 I}{4\pi R}\right)(-\hat{k}) = -\frac{2\mu_0 I}{4\pi R}\hat{k}.

  2. Semicircular arc: For a circular arc of radius RR subtending an angle θ\theta at the center, the field is B=μ0Iθ4πRB = \frac{\mu_0 I \theta}{4\pi R} . For a semicircle, θ=π\theta = \pi, so B=μ0Iπ4πRB = \frac{\mu_0 I \pi}{4\pi R}. Since the arc lies in the Y-Z plane, its axial field at the center OO will be along the X-axis. Given the current direction, the field points in the negative X-direction (−i^-\hat{i}). Field from arc = −πμ0I4πRi^-\frac{\pi\mu_0 I}{4\pi R}\hat{i}.

  3. Total Field: Summing the vectors: B⃗=−πμ0I4πRi^−2μ0I4πRk^=−μ0I4πR(πi^+2k^)\vec{B} = -\frac{\pi\mu_0 I}{4\pi R}\hat{i} - \frac{2\mu_0 I}{4\pi R}\hat{k} = -\frac{\mu_0 I}{4\pi R}(\pi\hat{i} + 2\hat{k}). This matches Option 3.

Exam Context & Concepts Covered

This question aligns with the NEET PHYSICS syllabus, specifically targeting concepts from MOVING CHARGES AND MAGNETISM. Mastering this topic is crucial for scoring well in the upcoming medical entrance examinations. Solving conceptually related problems will help you understand the nuances of these concepts and improve your problem-solving speed.

PHYSICSMOVING CHARGES AND MAGNETISMcarryingcurrentadjoiningfigurelinear

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