To tackle this problem, we need to understand the concept of magnetic dipole moments and how they relate to magnetic induction at specific points in space. The dipole moment of a magnet is a measure of its strength and orientation, and it plays a crucial role in determining the magnetic field produced by the magnet. In this case, we have a semicircular magnet with a dipole moment of 28 A·m², which is then transformed into a bar magnet and subsequently cut into two equal parts. Let's break this down step by step.
Understanding the Magnetic Dipole Moment
The dipole moment (p) of a magnet is defined as the product of the strength of the magnet (m) and the distance (d) between the poles. For a semicircular magnet, when it is straightened into a bar magnet, the dipole moment remains the same, but the configuration changes. When we cut the bar magnet along its axial line, we essentially create two magnets, each with half the dipole moment of the original.
Calculating the Magnetic Induction
To find the magnetic induction (B) at a point along the equatorial line and the axial line, we can use the formula for the magnetic field due to a dipole:
- For a point on the axial line: B = (μ₀ / 4π) * (2p / r³)
- For a point on the equatorial line: B = (μ₀ / 4π) * (p / r³)
Here, μ₀ is the permeability of free space (approximately 4π × 10⁻⁷ T·m/A), p is the dipole moment, and r is the distance from the center of the magnet to the point where we want to find the magnetic induction.
Step-by-Step Calculation
1. **Determine the new dipole moment**: After cutting the bar magnet, each piece will have a dipole moment of:
p' = p / 2 = 28 A·m² / 2 = 14 A·m²
2. **Calculate the magnetic induction at the equatorial line**: For a point 0.2 m away from the center:
B_equatorial = (μ₀ / 4π) * (p' / r³) = (4π × 10⁻⁷ / 4π) * (14 / (0.2)³)
B_equatorial = (10⁻⁷) * (14 / 0.008) = (10⁻⁷) * 1750 = 1.75 × 10⁻⁴ T
3. **Calculate the magnetic induction along the axial line**: Using the same distance:
B_axial = (μ₀ / 4π) * (2p' / r³) = (4π × 10⁻⁷ / 4π) * (2 * 14 / (0.2)³)
B_axial = (10⁻⁷) * (28 / 0.008) = (10⁻⁷) * 3500 = 3.5 × 10⁻⁴ T
Final Results
In summary, the magnetic induction at a point 0.2 m away from the center of the magnets is:
- Equatorial Line: B = 1.75 × 10⁻⁴ T
- Axial Line: B = 3.5 × 10⁻⁴ T
This analysis illustrates how the magnetic field varies depending on the position relative to the magnet and highlights the importance of understanding dipole moments in magnetism. If you have any further questions or need clarification on any part of this process, feel free to ask!
