Find the magnetic field at the point P in the figure. The curved… — Solution | JEE Physics

MCQMediumJEE 2023Biot–Savart Law

Find the magnetic field at the point $P$ in the figure. The curved portion is a semicircle connected to two long straight wires.

Figure showing a semicircular curved wire connected to a long horizontal wire at the top and a long vertical wire downward, with current $$I$$ marked and point $$P$$ near the center.

Duplicate overlay of the same wire figure showing the semicircular arc, two straight wires, current direction $$I$$, and point $$P$$ inside the semicircle.

A
$\frac{\mu_0 I}{2r} \left( 1 + \frac{2}{\pi} \right)$
B
$\frac{\mu_0 I}{2r} \left( 1 + \frac{1}{\pi} \right)$
C
$\frac{\mu_0 I}{2r} \left( \frac{1}{2} + \frac{1}{2\pi} \right)$
D
$\frac{\mu_0 I}{2r} \left( 1 + \frac{1}{\pi} \right)$

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: A semicircular curved wire of radius $r$ is connected to two long straight wires carrying current $I$ . The magnetic field is required at point $P$ .

Find: Magnetic field at $P$ .

Applying Biot-Savart’s Law:

B_p = \left( \frac{\mu_0 I}{4r} + \frac{\mu_0 I}{4\pi r} \right) = \frac{\mu_0 I}{2r} \left( \frac{1}{2} + \frac{1}{2\pi} \right)

Therefore, the magnetic field at $P$ is $\frac{\mu_0 I}{2r} \left( \frac{1}{2} + \frac{1}{2\pi} \right)$ . This matches option C. The solution labels option D, but the worked expression matches option C, so the worked expression is taken as authoritative.

Common mistakes

Adding the option label from the page instead of checking the worked expression is incorrect. The final formula from the solution must be matched with the options; here it matches option C, not the listed label D.
Using the magnetic field formula for a full circle instead of a semicircle is incorrect. For the curved part, use the semicircular contribution shown in the working, not the full circular field.
Ignoring the contribution of the straight wire segments is incorrect. The total field at $P$ is obtained by adding the contributions written in the solution expression.

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Answer

Step-by-step solution

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