What is the scientific relationship between the size of a neodymium magnet ring and its magnetic strength?

1. Basic definition of magnetic strength
Surface magnetic field strength (unit: Gauss or Tesla): indicates the size of the magnetic field on the surface of the magnet, which directly affects the adsorption force or the force on the external object.
Magnetic flux (unit: Weber): related to the volume of the magnet. The larger the volume, the higher the total magnetic flux.
Magnetic energy product (BHmax): a parameter that measures the energy storage capacity of the magnet, which is determined by the remanence (Br) and coercive force (Hc) of the material itself.

Neodymium Ring Magnet

2. The influence of size parameters on magnetic force
Outer diameter (OD) and inner diameter (ID):
Increase in outer diameter: increase the cross-sectional area of ​​the magnet (circular cross-sectional area = π×(OD² - ID²)/4), the total magnetic flux increases accordingly, but the surface magnetic field strength may be slightly reduced due to the diffusion of the magnetic field distribution.
Increase in inner diameter: Under the same outer diameter, the increase in inner diameter will reduce the volume of the magnet, resulting in a decrease in the total magnetic flux, but the magnetic field in the central area may be more concentrated (for example, when axially magnetized).
Thickness (height):
Increasing the thickness will directly increase the volume of the magnet, thereby increasing the total magnetic flux. However, the surface magnetic field strength does not increase linearly, because the attenuation of the magnetic field is inversely proportional to the square of the distance, and excessive thickness may cause the magnetic field distribution to be more dispersed.

3. Magnetization direction and magnetic field distribution
Axial magnetization (magnetic field along the thickness direction of the ring):
The magnetic field is concentrated at both ends of the ring (top and bottom surfaces), and the magnetic field in the center hole area is weak. Increasing the thickness will extend the magnetic field path and may slightly reduce the surface magnetic field strength.
Radial magnetization (magnetic field along the circumference of the ring):
The magnetic field is concentrated on the inner and outer diameter surfaces of the ring. At this time, the size difference between the inner diameter and the outer diameter will affect the uniformity of the magnetic field, and a smaller inner diameter may lead to a stronger internal magnetic field concentration.

4. Demagnetizing Field Effect (Demagnetizing Field)
The reverse magnetic field generated by the shape of the magnet itself weakens the effective magnetic field strength.
The demagnetization factor of a ring magnet is related to its aspect ratio (thickness/diameter). Thinner magnetic rings have stronger demagnetization fields, which may cause the actual magnetic force to be lower than the theoretical value; thicker magnetic rings have a weaker demagnetization effect, and the magnetic force is closer to the theoretical performance of the material.

5. Mathematical model and empirical law
Magnetic flux formula: total magnetic flux Φ ≈ Br × A (A is the cross-sectional area), indicating that the outer diameter and inner diameter indirectly determine the magnetic flux by affecting the cross-sectional area.
Surface magnetic field strength estimation: For axially magnetized ring magnets, the surface magnetic field strength (B) approaches the remanence (Br) as the thickness increases, but affected by the demagnetization field, the actual value is usually 50%~80% of Br.
Size limit: When the magnet size is too small (such as micro rings), the grain boundary effect of the material and the processing accuracy may cause a significant decrease in magnetic properties.

6. Trade-offs in practical applications
Motors and generators: high magnetic flux is required, and magnetic rings with larger outer diameters and thicknesses are usually selected, but space limitations and eddy current losses must be considered.
Sensors and magnetic coupling: relying on high surface magnetic field strength, smaller inner diameters and thinner magnetic rings may be selected to concentrate the magnetic field.
Magnetic adsorption: The total magnetic flux (adsorption force) and the magnetic field gradient (action distance) need to be balanced. For example, increasing the thickness can extend the adsorption distance, but it needs to be optimized with the magnetic conductive material.

7. Experimental verification case
Outer diameter is fixed, inner diameter changes: the inner diameter increases from 5mm to 15mm (outer diameter 30mm), the total magnetic flux decreases by about 40%, but the magnetic field strength in the central area increases by 20% (axial magnetization).
Thickness doubled: the thickness increases from 5mm to 10mm (outer diameter 20mm, inner diameter 10mm), the surface magnetic field strength increases from 4500 Gauss to 6000 Gauss, but when it continues to increase to 15mm, it only increases to 6300 Gauss, and the increase rate slows down.