A toroid of n turns, mean radius R and cross-sectional radius a carries current I. It is placed on a horizontal table taken as x-y plane. Its magnetic moment `vecM`

Options

A. is non-zero and points in the Z-direaction by symmetry
B. points along the axis of the toroid `( m = m phi)`
C. is zero, otherwise there would be a field falling as `(1)/(r^(3))` at large distance outside the toriod
D. is pointing radially outwards

Explanation

To find the magnetic moment \(\vec{M}\) of a toroid with \(n\) turns, mean radius \(R\), cross-sectional radius \(a\), and carrying current \(I\), we can follow these steps:

### Step 1: Understand the Geometry of the Toroid
A toroid can be visualized as a solenoid bent into a circular shape. The mean radius \(R\) is the average distance from the center of the toroid to the wire, while \(a\) is the radius of the cross-section of the toroid.

### Step 2: Formula for Magnetic Moment
The magnetic moment \(\vec{M}\) for a toroid is given by the formula:
\[ \vec{M} = n \cdot I \cdot A \]
where:
- \(n\) is the number of turns,
- \(I\) is the current flowing through the toroid,
- \(A\) is the area of the cross-section of the toroid.

### Step 3: Calculate the Cross-sectional Area
The cross-sectional area \(A\) of the toroid can be calculated as:
\[ A = \pi a^2 \]
where \(a\) is the radius of the cross-section.

### Step 4: Substitute the Area into the Magnetic Moment Formula
Now, substituting the area \(A\) into the magnetic moment formula:
\[ \vec{M} = n \cdot I \cdot (\pi a^2) \]

### Step 5: Analyze the Magnetic Moment
The direction of the magnetic moment \(\vec{M}\) is along the axis of the toroid. However, since the magnetic field is confined within the toroid and there is no magnetic field outside, we can conclude that the magnetic moment is effectively zero in the regions outside the toroid.

### Step 6: Conclusion
Thus, the magnetic moment \(\vec{M}\) exists within the toroid but is zero in the surrounding space. Therefore, the magnetic moment is:
\[ \vec{M} = n \cdot I \cdot \pi a^2 \]
and it is confined to the interior of the toroid.