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Question #42780

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The Question

A long straight wire carrying a steady current $I$ along the positive $z$ -axis is placed in a vacuum (permeability $\mu_0$ ). A uniform external magnetic field $\vec{B}_0 = B_0 \hat{j}$ is also present throughout the space. Consider a cylindrical region of volume $V$ defined by the coordinates $a \le r \le R$ , $0 \le \theta \le \pi/2$ , and $0 \le z \le L$ , where $(r, \theta, z)$ are standard cylindrical coordinates with $x = r\cos\theta$ and $y = r\sin\theta$ . What is the total magnetic energy stored within this volume $V$ ?

Options

A.
$\frac{\mu_0 I^2 L}{16\pi} \ln\left(\frac{R}{a}\right) + \frac{\pi B_0^2 L}{8\mu_0} (R^2 - a^2) + \frac{I B_0 L}{2\pi} (R - a)$
B.
$\frac{\mu_0 I^2 L}{16\pi} \ln\left(\frac{R}{a}\right) + \frac{\pi B_0^2 L}{8\mu_0} (R^2 - a^2)$
C.
$\frac{\mu_0 I^2 L}{16\pi} \ln\left(\frac{R}{a}\right) + \frac{\pi B_0^2 L}{8\mu_0} (R^2 - a^2) - \frac{I B_0 L}{2\pi} (R - a)$
D.
$\frac{\mu_0 I^2 L}{8\pi} \ln\left(\frac{R}{a}\right) + \frac{\pi B_0^2 L}{4\mu_0} (R^2 - a^2) + \frac{I B_0 L}{\pi} (R - a)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Energy Density Locality|Gauss Law Applicability|Superposition of Magnetic Fields

Why This Question Is Tricky

Common Mistakes

Students incorrectly assume that the total magnetic energy of a system is simply the sum of the individual self-energies of the constituent fields ( $U = U_1 + U_2$ ).

They fail to recognize that energy density is proportional to the square of the net field ( $u \propto |\vec{B}_{net}|^2$ ), which introduces a non-linear cross-term $\vec{B}_1 \cdot \vec{B}_2$ .

While this interaction energy often integrates to zero over all space due to symmetry, restricting the integration volume to a specific quadrant prevents the angular integral from vanishing.

Ignoring this 'locality' constraint of energy density leads to missing the interaction term entirely.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Energy Density Locality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free