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Question #42902

Difficulty: 8/10 | 🟢 Free

The Question

A tightly wound, ideal semi-infinite solenoid of radius $R$ lies along the negative $z$ -axis ( $z \le 0$ ). It carries a steady current $I$ and has $n$ turns per unit length. Consider two open surfaces:\n $S_1$ : An annular flat disk on the $z = 0$ plane, defined by $R \le \sqrt{x^2+y^2} \le 2R$ .\n $S_2$ : A hemispherical shell defined by $x^2 + y^2 + z^2 = 4R^2$ with $z \ge 0$ .\nLet $\Phi_1$ be the magnetic flux passing in the $+z$ direction through $S_1$ , and $\Phi_2$ be the magnetic flux passing radially outward through $S_2$ . Which of the following represents the correct values of $\Phi_1$ and $\Phi_2$ ?

Options

A.
$\Phi_1 = 0$ , $\Phi_2 = \frac{1}{2} \mu_0 n I \pi R^2$
B.
$\Phi_1 = \frac{1}{4} \mu_0 n I \pi R^2$ , $\Phi_2 = \frac{3}{4} \mu_0 n I \pi R^2$
C.
$\Phi_1 = 0$ , $\Phi_2 = \mu_0 n I \pi R^2$
D.
$\Phi_1 = \frac{1}{2} \mu_0 n I \pi R^2$ , $\Phi_2 = \mu_0 n I \pi R^2$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Electric Field Line Geometry|Solenoid Fringe Fields|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the misconception that magnetic field lines 'spreading out' from the end of a solenoid implies a non-zero axial field just outside the solenoid's rim.

Students intuitively assume that fringe fields must penetrate the annular region $S_1$ , failing to realize that at the exact plane of symmetry ( $z=0$ ), the field is purely radial outside the solenoid.

Furthermore, it tests the application of Gauss's Law for Magnetism ( $\nabla \cdot \mathbf{B} = 0$ ) to relate the flux through a complex curved surface ( $S_2$ ) to a simple flat surface, requiring the correct piecewise definition of the $B_z$ component derived via the superposition principle.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Electric Field Line Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free