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NEET PHYSICSMOVING CHARGES AND MAGNETISMMedium

Question

A very long conducting wire is bent in a semi-circular shape from A to B as shown in the figure. The magnetic field at the point P for steady current configuration is given by:

A

\dfrac{\mu_0 i}{4R}\left[1-\dfrac{2}{\pi}\right] \text{ pointed into the page}

B

\dfrac{\mu_0 i}{4R} \text{ pointed into the page}

C

\dfrac{\mu_0 i}{4R} \text{ pointed away from the page}

D

\dfrac{\mu_0 i}{4R}\left[1-\dfrac{2}{\pi}\right] \text{ pointed away from the page}

Step-by-Step Solution

  1. Analyze the Geometry: The wire consists of three segments: two semi-infinite straight wires and one semi-circular arc. Based on the answer format (1−2/π1 - 2/\pi), the fields from the straight segments must oppose the field from the arc.
  2. Field due to Straight Segments: Consider the two semi-infinite straight wires extending to infinity. The magnetic field B1B_1 due to a semi-infinite wire at a perpendicular distance RR from one end is given by B1=μ0I4πRB_1 = \frac{\mu_0 I}{4\pi R}. Since there are two such wires carrying current in directions that produce magnetic fields in the same direction (opposing the arc), their combined field is: Bstraight=2×μ0I4πR=μ0I2πRB_{straight} = 2 \times \frac{\mu_0 I}{4\pi R} = \frac{\mu_0 I}{2\pi R}
  3. Field due to Semi-Circular Arc: The magnetic field BarcB_{arc} at the center of a semi-circular arc of radius RR is half that of a full circle: Barc=12(μ0I2R)=μ0I4RB_{arc} = \frac{1}{2} \left( \frac{\mu_0 I}{2R} \right) = \frac{\mu_0 I}{4R}
  4. Net Magnetic Field: The net magnetic field is the difference between the arc field and the straight wire field (assuming the standard configuration where they oppose): Bnet=Barc−Bstraight=μ0I4R−μ0I2πRB_{net} = B_{arc} - B_{straight} = \frac{\mu_0 I}{4R} - \frac{\mu_0 I}{2\pi R} Factor out μ0I4R\frac{\mu_0 I}{4R}: Bnet=μ0I4R(1−2π)B_{net} = \frac{\mu_0 I}{4R} \left( 1 - \frac{2}{\pi} \right)
  5. Direction: The term 11 (from the arc) is greater than 2π≈0.63\frac{2}{\pi} \approx 0.63 (from the wires). Therefore, the net field direction is determined by the arc. If the arc current is counter-clockwise, the field points out of the page (away). The option 'pointed away from the page' corresponds to this standard case.

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 MAGNETISMconductingsemicircularfiguremagneticsteady

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