Calculator of force between a cylindrical magnet and a paper clip
 | Stan Zurek, Calculator of force between a cylindrical magnet and a paper clip, Encyclopedia Magnetica, https://www.e-magnetica.pl/doku.php/calculator/force_between_magnet_and_clip, {accessed: 2025-03-13} |
 See more: Magnets in conservation, restoration, and display of art |
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This calculator estimates the level of magnetic force generated by a cylindrical magnet on a paper clip made from a magnetic steel wire.
Equations
| Source: This calculator was proposed by S. Zurek (unpublished elsewhere) | ||
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| Other sources: [1] F. Fiorillo, Measurement and Characterization of Magnetic Materials, Academic Press, 2005, ISBN 9780122572517 [2] E.P. Furlani, Permanent magnet and electromechanical devices, Academic Press, London, 2021, ISBN 0122699513 [3] B.D. Cullity, C.D. Graham, Introduction to Magnetic Materials, 2nd edition, Wiley, IEEE Press, New Jersey, 2009, ISBN 9780471477419 |
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| demagnetising factor of a single cylindrical wire based on [1] Fiorillo, eq. (1.20), p. 11 | $$ N_d = \frac{1}{2 ⋅ \sqrt{1+\frac{d^2}{D^{~2}}}} $$ | (unitless) |
| magnetic field strength at the distance x from the surface of a cylindrical magnet based on [2] Furlani, eq. (3.98), p. 129 | $$ H(x) = \frac{B_r}{2 ⋅ μ_0} ⋅ \left( \frac{x+L}{\frac{D^{~2}}{4} + (x+L)^2} - \frac{x}{\frac{D^{~2}}{4} + x^2} \right) $$ | (A/m) |
| gradient of magnetic field across the first half of the wire thickness (from wire start to wire centre) | $$ \frac{dH}{dx} = \frac{H_{start} - H_{centre}}{(x=t) - (x= t+d/2)} = \frac{ H(t) - H(t+d/2) }{d/2} $$ | (A/m) |
| magnetic field strength at the centre of the magnetic wire [2] Furlani, eq. (1.46)-(1.47), p. 24 | $$ H_{int} = \frac{ H_{centre} }{1 + N_d ⋅ (μ_r - 1) } $$ | (A/m) |
| magnetisation at the centre of the magnetic wire [2] Furlani, eq. (1.51), p. 26 | $$ M_{int} = (μ_r - 1) ⋅ H_{int} $$ | (A/m) |
| Volume V of a single wire (estimated across the diameter of the magnet) | $$ V_{wire} = \frac{π}{4} ⋅ d^2 ⋅ D $$ | (m3) |
| Magnetic moment m of a single wire [3] Cullity & Graham, eq. (2.60), p. 81 | $$ m_{wire} = M_{int} ⋅ V_{wire} $$ | (A⋅m2) |
| Total force Ftot acting on n = 4 wires [3] Cullity & Graham, eq. (2.60), p. 81 | $$ F_{tot} = m_{wire} ⋅ \frac{dH}{dx} ⋅ μ_0 ⋅ n $$ | (N) |
| Note: This calculator is based on the magnetic field strength along the axis of magnet, and therefore does not take into account the field non-uniformity. An assumption is also made that the straight parts of the paper clip are fully overlapping the magnet. The calculated force can differ significantly from that occurring in reality, especially if other magnetic components are present in the vicinity of the magnet. | ||
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