A conducting wire carrying a steady current \(I\) is shaped as shown in the figure below. All connected straight segments meet at right angles. What is the magnetic moment of the current loop?
Show Hint
To find the area vectors of non-planar loops quickly, project the loop onto the principal planes (\(xy\), \(yz\), \(xz\)).
The magnetic moment vector is simply the sum of the moments of these projected 2D loops.
To find the magnetic moment of the current loop, we first analyze the shape of the wire. The loop forms an L-shaped structure in a 3D plane. The task is to find the magnetic moment of the given structure.
The magnetic moment \(\vec{m}\) of a current-carrying loop is given by:
\[\vec{m} = I \cdot \vec{A}\]
where \(\vec{A}\) is the area vector perpendicular to the plane of the loop. For a loop composed of two rectangular planes intersecting at right angles, we can calculate the area vector for each part separately and then add them vectorially.
In the image, the loop consists of two rectangles, each contributing to the magnetic moment:
The rectangle in the \(xy\)-plane has an area:
\[A_{xy} = a \times b\]
The area vector points in the \(\hat{k}\) direction.
The rectangle in the \(yz\)-plane has an area:
\[A_{yz} = a \times b\]
The area vector points in the \(-\hat{j}\) direction due to the right-hand rule.
The total area vector is:
\[\vec{A} = a b \hat{k} - a b \hat{j}\]
Now, calculate the magnetic moment:
\[\vec{m} = I \vec{A} = I a b (\hat{k} - \hat{j})\]
Thus, the magnetic moment of the loop is:
\[I a b (\hat{k} - \hat{j})\]
This matches with the given correct answer option: \(I a b (\hat{k} - \hat{j})\).
