Question

If a magnet of pole strength \(\mathrm{m}\) is divided into four parts such that the length and width of each part is half that of initial one, then the pole strength of each part will be (a) \((\mathrm{m} / 4)\) (b) \((\mathrm{m} / 2)\) (c) \((\mathrm{m} / 8)\) (d) \(4 \mathrm{~m}\)

Short answer

The pole strength of each part will be (a) \(\frac{m}{4}\).

Step-by-step solution

Understand the concept of pole strength

Pole strength is a measure of the strength of a magnetic pole in a magnet. It is directly proportional to the product of the magnet's length, width and thickness. In our problem, we have a magnet with a given pole strength m, and we are asked to find the pole strength of each of the four parts after dividing the magnet.

Determine the relationship between the pole strength and size of the magnet

Let the lengths of the original magnet be L (length), W (width), and the pole strength be m. When the magnet is divided into four parts, the dimensions of each part will be L/2 (length) and W/2 (width).

Calculate the pole strength of each part

Since the pole strength is proportional to the product of the magnet's length, width and thickness, we can express the pole strength of a single part (m') as: \(m' \propto LW\) Where L and W are the length and width of a single part respectively. Now, let's substitute the dimensions of the smaller magnets: \(m' \propto \frac{L}{2} \times \frac{W}{2}\) \(m' \propto \frac{LW}{4}\) Since the proportionality constant is the same for both the original magnet and the smaller pieces, we can write: \(\frac{m'}{m} = \frac{\frac{LW}{4}}{LW}\) By simplifying the equation, we get: \(m' = \frac{m}{4}\)

Choose the correct answer

Comparing our result with the given options, we can see that the pole strength of each part is: (a) \(\frac{m}{4}\) This is in alignment with our calculated result in Step 3. Therefore, the correct answer is (a). Each part of the divided magnet will have a pole strength of \(\frac{m}{4}\).

Key concepts

Magnet Division

When a magnet is divided into smaller parts, it's important to understand how the magnetic properties change. In the case of our exercise, the magnet is split into four distinct parts. This division involves altering the magnet's physical dimensions such as length and width. In our scenario, both the length and width of the magnet are halved once divided.

This change in size directly affects the pole strength of each resulting part. Pole strength is the ability of a magnet's pole to attract or repel other magnets or magnetic materials. After the division, we need to consider how this strength is distributed among the smaller pieces formed.

Proportionality Relationship

The concept of proportionality plays a key role in magnetism and, more specifically, in calculating pole strength. In this context, pole strength is directly proportional to specific dimensions of a magnet—the length, width, and thickness. If any of these dimensions change, the pole strength will change in a proportional manner.

In our example, the magnet is divided into smaller pieces with each having new dimensions. Since pole strength is related to the product of length, width, and thickness, any change in these dimensions will result in a change in pole strength proportionally. This allows us to calculate how the original magnet's strength is split among the new, smaller magnets.

Magnetic Dimensions

The physical dimensions of a magnet—length, width, and thickness—play a crucial role in defining its magnetic properties. In our scenario, we begin with a magnet with specific dimensions and alter them through division.

When the magnet is partitioned into smaller parts where the length and width both drop to half of the original, these new dimensions become pivotal in recalculating the pole strength. By knowing how dimension changes, you can correlate them directly to changes in magnetic properties of the magnet parts.

Calculation of Pole Strength

Determining the pole strength of a magnetic section after it has been divided involves understanding the relationship between its dimensions and magnetic properties. As shown in the solution, the pole strength varies with the product of the length and width.

For our problem, after reducing both length and width by half, we utilize this understanding to compute the new pole strength. This calculation reveals that the pole strength of each of the four parts results in being a quarter (\( \frac{m}{4} \)) of the original magnet. This stems from the proportional relationship between dimensional changes and resultant magnetic strength, allowing us to predict outcomes in practical scenarios.