Three magnets of same length but moments `M,2M` and `3M` are arranged in the form of an equilateral triangle with oppositive poles nearer, the resultnat magnetic moment of the arrangement is

Options

A. `6M`
B. `zero`
C. `sqrt3M`
D. `(sqrt3)/(2)M`

Explanation

To find the resultant magnetic moment of three magnets arranged in an equilateral triangle with moments \(M\), \(2M\), and \(3M\), we can follow these steps:

### Step 1: Understand the Arrangement
We have three magnets with magnetic moments \(M\), \(2M\), and \(3M\) arranged in an equilateral triangle. The magnets are positioned such that opposite poles are closer together.

### Step 2: Calculate the Pole Strengths
The magnetic moment \(m\) of a magnet is given by the formula:
\[ m = p \cdot L \]
where \(p\) is the pole strength and \(L\) is the length of the magnet. Since the lengths of the magnets are the same, we can express the pole strengths as:
- For magnet with moment \(M\): \(p_1 = \frac{M}{L}\)
- For magnet with moment \(2M\): \(p_2 = \frac{2M}{L}\)
- For magnet with moment \(3M\): \(p_3 = \frac{3M}{L}\)

### Step 3: Determine the Net Pole Strength
Since the magnets are arranged with opposite poles closer, we need to consider the net pole strength. The net pole strength can be calculated by subtracting the pole strengths of the magnets:
\[ \text{Net Pole Strength} = p_3 - p_1 - p_2 = \frac{3M}{L} - \frac{M}{L} - \frac{2M}{L} \]
This simplifies to:
\[ \text{Net Pole Strength} = \frac{3M - M - 2M}{L} = \frac{0}{L} = 0 \]
This means that the net effect of the pole strengths cancels out.

### Step 4: Calculate Resultant Magnetic Moment
Since the net pole strength is zero, we need to consider the arrangement of the magnets. The effective magnetic moment can be calculated using vector addition, as the magnets are not aligned in a straight line but in an equilateral triangle.

Using the parallelogram law of vector addition for the two resultant magnetic moments:
Let \(m_1 = M\) and \(m_2 = 2M\), and the angle between them is \(60^\circ\):
\[ R = \sqrt{m_1^2 + m_2^2 + 2m_1m_2 \cos(60^\circ)} \]
Substituting the values:
\[ R = \sqrt{M^2 + (2M)^2 + 2 \cdot M \cdot 2M \cdot \frac{1}{2}} \]
\[ = \sqrt{M^2 + 4M^2 + 2M^2} = \sqrt{7M^2} = \sqrt{7}M \]

### Step 5: Final Result
The resultant magnetic moment of the arrangement is:
\[ R = \sqrt{7}M \]