Flux density can be illustrated the density per unit area of lines of magnetic force. Higher magnetic flux density corresponds to more closely spaced lines of force. Magnet geometry affects relevant variables and equation used to calculate flux density. The air gap, or distance, refers to distance from magnet surface at which flux density is measured. Formula and calculation for a rod-shaped magnet incorporate magnet radius, thickness, and air gap.

- Skill level:
- Easy

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### Things you need

- Calculator with exponent capacity
- Magnet and magnet dimension data
- Air gap distance data

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## Instructions

- 1
Incorporate magnet geometry. MagneticSolution's “Flux Density Formulae” points out a consistent theme—length (L), width (W), and thickness (T) affect magnetic flux. If the magnet has a circular component, then magnet radius (R) also comes into play. Magnetic lines of force that are close together correspond to higher flux density. If magnet geometry more easily allows more spacing between magnetic lines of force, magnetic flux intensity will drop off quicker with increasing distance.

- 2
Factor in the air gap. Flux density can be some distance away from the magnet, or at the magnet surface. If at the surface, distance (x) is zero, x = 0. This can simplify calculations. Magnetic flux density tends to decrease as air gap increases.

- 3
Program or write out the formula for specific magnet geometry. For example, the formula for a single steel-backed rod magnet can be broken down in terms of geometric variables. Rod magnet radius and thickness are relevant in flux density calculations. Note that rod length does not affect flux density calculation. Due to formula complexity, secondary substitution is necessary. Australian Magnetic Solutions has a list of flux density formulas. A single magnetic rod with specified radius (R) and thickness (T) has flux density distance x from the magnet’s axis. The following equation describes this flux density:

B = (Br/2) * ([A/C] – [x/B]). A = x + 2T, B = sqrt(R^2 + x^2), C = sqrt(R^2 + A^2).

The “Br” is the magnetic flux constant. Br relates to field strength within the magnet substance. Obviously, a weak magnet gives less outside flux density than a strong magnet with the same geometry and distance.

This formula is innately complex and difficult to express in text format without secondary substitution such as “A = x+2T.”

- 4
Use the previous equation to calculate flux density with hypothetical numbers substituted into the equation. Assume that the following values are given:

Br = 2kg

R = 6.0cm, T = 0.5cm, x = 3.0mm

First, we convert 2kg (two kilogauss) into standard gauss units: 2 kG = 2000 Gauss (2000 G).

Next, make all length units the same. Express x in centimetres (cm) as x = 0.3cm.

Calculate A = x + 2T = 0.3 + (2*0.5) = 0.3 + 1 = 1.3 cm

Similarly, calculate B = sqrt(6^2 + 0.3^2) = sqrt(36 + 0.09) = sqrt(36.09) = 6.007cm

Find C, C = sqrt(6^2 + 1.3^2) = sqrt(36 + 1.69) = sqrt(37.69) = 6.139cm

Substitute A, B, C, and Br values into overall formula:

B = (2000/2) * ((1.3/6.139) – (0.3/6.007)) = 1000 * (0.2117 – 0.04994) = 161.76 Gauss.

#### Tips and warnings

- In this example, significant digits were not considered. In real life situations, precision limits and significant digits would affect the final answer. For instance, the value 161.76 Gauss would be rounded to either 161.8 or 162 (or 160), depending on precision of the measuring and calculating devices.