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-direction by symmetry
B. points along the axis of the toroid `(vec(M)=mhatphi)`
C. is zero, otherwise there would be a field falling as `(1)/(r^(3))` at large distances outside the toroid
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-by-Step Solution:

1. **Understanding the Geometry of the Toroid**:
- A toroid is essentially a circular ring with wire wound around it. The mean radius \( R \) is the distance from the center of the toroid to the center of the wire coil, while \( a \) is the radius of the cross-section of the toroid.

2. **Magnetic Moment Formula**:
- The magnetic moment \( \vec{M} \) of a current-carrying loop is given by the formula:
\[ \vec{M} = n \cdot I \cdot A \]
where \( n \) is the number of turns, \( I \) is the current, and \( A \) is the area of the loop.

3. **Calculating the Area \( A \)**:
- For a toroid, the area \( A \) can be approximated as the area of the circular cross-section of the toroid:
\[ A = \pi a^2 \]
- Here, \( a \) is the radius of the cross-section.

4. **Substituting the Area into the Magnetic Moment Formula**:
- Now substituting the expression for area \( A \) into the magnetic moment formula:
\[ \vec{M} = n \cdot I \cdot \pi a^2 \]

5. **Direction of the Magnetic Moment**:
- The direction of the magnetic moment \( \vec{M} \) is along the axis of the toroid, following the right-hand rule.

6. **Conclusion**:
- Therefore, the magnetic moment of the toroid is given by:
\[ \vec{M} = n \cdot I \cdot \pi a^2 \]
- It is important to note that the magnetic field is confined within the toroid, and outside the toroid, the magnetic field is zero.

### Final Answer:
The magnetic moment \( \vec{M} \) of the toroid is:
\[ \vec{M} = n \cdot I \cdot \pi a^2 \]