Homopolar Motor Animation

Introduction

            A homopolar motor has only three parts:

1. A cylindrical dry cell battery

2. A strong magnet positioned along the axis of the battery

3. A non-magnetic (e.g. copper) wire forming the armature

The wire is formed so that it conducts current from the top battery terminal to the bottom terminal by rubbing against the magnets which are in contact with the bottom terminal.  Figure 1 from the animation is a good diagram of the parts.

Figure 1: Diagram of the parts of the homopolar motor.  Note that the torque contribution Vs position on the armature is plotted.

Physics

            To compute the torque on the armature, the magnetic induction (field) B for an extended permanent magnet must first be computed.  This was done with the help of some integral equations from a Mathematica demonstration.  The radial, ρ, and axial, z, components of the field are found from the following integrals:

 

 

  

 

 

B z (ρ,z)=− ∫ 0 2π dϕ ∫ 0 a { r( L 2 −z ) [ r 2 + ( L 2 −z ) 2 + ρ 2 −2ρrcosϕ ] 3/2 + r( L 2 +z ) [ r 2 + ( L 2 +z ) 2 + ρ 2 −2ρrcosϕ ] 3/2 }dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkeadaWgaa WcbaGaamOEaaqabaGccaGGOaGaeqyWdiNaaiilaiaadQhacaGGPaGa eyypa0JaeyOeI0Yaa8qCaeaacaWGKbGaeqy1dygaleaacaaIWaaaba GaaGOmaiabec8aWbqdcqGHRiI8aOWaa8qCaeaadaGadaqaamaalaaa baGaamOCamaabmaabaWaaSaaaeaacaWGmbaabaGaaGOmaaaacqGHsi slcaWG6baacaGLOaGaayzkaaaabaWaamWaaeaacaWGYbWaaWbaaSqa beaacaaIYaaaaOGaey4kaSYaaeWaaeaadaWcaaqaaiaadYeaaeaaca aIYaaaaiabgkHiTiaadQhaaiaawIcacaGLPaaadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcqaHbpGCdaahaaWcbeqaaiaaikdaaaGccqGHsi slcaaIYaGaeqyWdiNaamOCaiGacogacaGGVbGaai4Caiabew9aMbGa ay5waiaaw2faamaaCaaaleqabaGaaG4maiaac+cacaaIYaaaaaaaki abgUcaRmaalaaabaGaamOCamaabmaabaWaaSaaaeaacaWGmbaabaGa aGOmaaaacqGHRaWkcaWG6baacaGLOaGaayzkaaaabaWaamWaaeaaca WGYbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaadaWcaaqa aiaadYeaaeaacaaIYaaaaiabgUcaRiaadQhaaiaawIcacaGLPaaada ahaaWcbeqaaiaaikdaaaGccqGHRaWkcqaHbpGCdaahaaWcbeqaaiaa ikdaaaGccqGHsislcaaIYaGaeqyWdiNaamOCaiGacogacaGGVbGaai 4Caiabew9aMbGaay5waiaaw2faamaaCaaaleqabaGaaG4maiaac+ca caaIYaaaaaaaaOGaay5Eaiaaw2haaiaadsgacaWGYbaaleaacaaIWa aabaGaamyyaaqdcqGHRiI8aaaa@8F7D@   (1.1)

B ρ (ρ,z)= ∫ 0 2π dϕ ∫ 0 a { r( ρ−rcosϕ ) [ r 2 + ( L 2 −z ) 2 + ρ 2 −2ρrcosϕ ] 3/2 − r( ρ−rcosϕ ) [ r 2 + ( L 2 +z ) 2 + ρ 2 −2ρrcosϕ ] 3/2 }dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkeadaWgaa WcbaGaeqyWdihabeaakiaacIcacqaHbpGCcaGGSaGaamOEaiaacMca cqGH9aqpdaWdXbqaaiaadsgacqaHvpGzaSqaaiaaicdaaeaacaaIYa GaeqiWdahaniabgUIiYdGcdaWdXbqaamaacmaabaWaaSaaaeaacaWG YbWaaeWaaeaacqaHbpGCcqGHsislcaWGYbGaci4yaiaac+gacaGGZb Gaeqy1dygacaGLOaGaayzkaaaabaWaamWaaeaacaWGYbWaaWbaaSqa beaacaaIYaaaaOGaey4kaSYaaeWaaeaadaWcaaqaaiaadYeaaeaaca aIYaaaaiabgkHiTiaadQhaaiaawIcacaGLPaaadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcqaHbpGCdaahaaWcbeqaaiaaikdaaaGccqGHsi slcaaIYaGaeqyWdiNaamOCaiGacogacaGGVbGaai4Caiabew9aMbGa ay5waiaaw2faamaaCaaaleqabaGaaG4maiaac+cacaaIYaaaaaaaki abgkHiTmaalaaabaGaamOCamaabmaabaGaeqyWdiNaeyOeI0IaamOC aiGacogacaGGVbGaai4Caiabew9aMbGaayjkaiaawMcaaaqaamaadm aabaGaamOCamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaabmaabaWa aSaaaeaacaWGmbaabaGaaGOmaaaacqGHRaWkcaWG6baacaGLOaGaay zkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeqyWdi3aaWbaaSqa beaacaaIYaaaaOGaeyOeI0IaaGOmaiabeg8aYjaadkhaciGGJbGaai 4BaiaacohacqaHvpGzaiaawUfacaGLDbaadaahaaWcbeqaaiaaioda caGGVaGaaGOmaaaaaaaakiaawUhacaGL9baacaWGKbGaamOCaaWcba GaaGimaaqaaiaadggaa0Gaey4kIipaaaa@98D3@      (1.2)

The contribution to the torque from a short element, dz. of the armature wire is:

δτ=i r a ×dl×B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaahs 8acqGH9aqpcaWGPbGaaCOCamaaBaaaleaacaWHHbaabeaakiaahEna caWHKbGaaCiBaiaahEnacaWHcbaaaa@425C@  

(1.3)

where ra is the radial vector from the magnet axis to the armature, i is the battery current, dl is the element of armature length, and B is the magnetic induction vector.

The total torque about the armature axis is twice the sum of all the contributions from equation (1.3).