mEC O1

Solution

Analysis:

We can use the Principle of Superposition to solve this problem by treating the path in Figure-II as a combination of current loops on the faces of the cube.

1. Analyze Figure-I:

• The current flows in a square loop on the front face (in the \(x-z\) plane).
• Using the Right-Hand Thumb Rule, the magnetic field direction at the center is along the positive \(y\)-axis (\(\hat{j}\)).
• Given magnitude: \(B_0\). So, \(\vec{B}_1 = B_0\hat{j}\).

2. Analyze Figure-II:

The complex path in Figure-II can be decomposed into three separate square loops on three faces of the cube:

1. Front Face (normal to \(y\)): The current flows in the same direction as Figure-I.
   Contribution: \(+B_0\hat{j}\).
2. Top Face (normal to \(z\)): The current flows counter-clockwise when viewed from above.
   Contribution: \(+B_0\hat{k}\).
3. Left Face (normal to \(x\)): The current flows in a way that generates a field pointing into the negative \(x\) direction.
   Contribution: \(-B_0\hat{i}\).

Conclusion:

The net magnetic field is the vector sum of these contributions:

$$ \vec{B}_{net} = -B_0\hat{i} + B_0\hat{j} + B_0\hat{k} $$ $$ \vec{B}_{net} = B_0(-\hat{i} + \hat{j} + \hat{k}) $$

Correct Answer: (c) \( B_0(-\hat{i} + \hat{j} + \hat{k}) \)