A long magnet is cut in two parts in such a way that the ratio of their lengths is 2:1. The retio of pole strengths of both the section is

Options

A. Equal
B. In the ratio of `2:1`
C. In the ratio of `1:2`
D. In the ratio of `4:1`

Answer

To solve the problem of finding the ratio of pole strengths of two sections of a long magnet that is cut in the ratio of 2:1, we can follow these steps:

### Step 1: Understand the Concept of Pole Strength
Pole strength (denoted as \( p \)) is a measure of the strength of a magnetic pole to attract magnetic materials. It is defined as the magnetic moment per unit length of the magnet.

### Step 2: Define the Lengths of the Sections
Let the original length of the magnet be \( L \). When the magnet is cut into two parts in the ratio of 2:1, we can denote the lengths of the two sections as:
- Length of the first section (longer part) = \( \frac{2L}{3} \)
- Length of the second section (shorter part) = \( \frac{L}{3} \)

### Step 3: Understand the Magnetic Moment
The magnetic moment (\( M \)) of a magnet is given by the product of its pole strength and its length:
\[ M = p \times L \]
Where \( p \) is the pole strength and \( L \) is the length of the magnet.

### Step 4: Analyze the Magnetic Moments of Each Section
Let \( p_1 \) be the pole strength of the first section and \( p_2 \) be the pole strength of the second section. The magnetic moments for the two sections can be expressed as:
- For the first section:
\[ M_1 = p_1 \times \frac{2L}{3} \]
- For the second section:
\[ M_2 = p_2 \times \frac{L}{3} \]

### Step 5: Relate the Magnetic Moments
Since the original magnet is not losing any magnetic moment when cut, we can equate the total magnetic moment before and after cutting:
\[ M = M_1 + M_2 \]
However, since we are interested in the ratio of pole strengths and we know that the total magnetic moment remains constant, we can analyze the relationship between the pole strengths.

### Step 6: Calculate the Ratio of Pole Strengths
From the magnetic moment equations, we can express the ratio of the pole strengths:
\[ \frac{M_1}{M_2} = \frac{p_1 \times \frac{2L}{3}}{p_2 \times \frac{L}{3}} = \frac{p_1 \times 2}{p_2} \]
Since the total magnetic moment remains constant, we can conclude that:
\[ p_1 \times \frac{2L}{3} + p_2 \times \frac{L}{3} = M \]
This implies that the pole strengths are directly proportional to the lengths of the sections, thus:
\[ \frac{p_1}{p_2} = \frac{2}{1} \]

### Conclusion
The ratio of pole strengths of the two sections is:
\[ \frac{p_1}{p_2} = 2:1 \]