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</style><table class="sidebar sidebar-collapse nomobile nowraplinks em-sidebar"><tbody><tr><th class="sidebar-title"><a href="Electromagnetism" title="Electromagnetism">Electromagnetism</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Electricity" title="Electricity">Electricity</a></li>
<li><a href="Magnetism" title="Magnetism">Magnetism</a></li>
<li><a href="Optics" title="Optics">Optics</a></li>
<li><a href="History_of_electromagnetic_theory" title="History of electromagnetic theory">History</a></li>
<li><a href="Computational_electromagnetics" title="Computational electromagnetics">Computational</a></li>
<li><a href="List_of_textbooks_in_electromagnetism" title="List of textbooks in electromagnetism">Textbooks</a></li>
<li><a href="List_of_electrical_phenomena" title="List of electrical phenomena">Phenomena</a></li></ul></td>
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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Electrostatics" title="Electrostatics">Electrostatics</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Charge_density" title="Charge density">Charge density</a></li>
<li><a href="Electrical_conductor" title="Electrical conductor">Conductor</a></li>
<li><a href="Coulomb's_law" title="Coulomb's law">Coulomb law</a></li>
<li><a href="Electret" title="Electret">Electret</a></li>
<li><a href="Electric_charge" title="Electric charge">Electric charge</a></li>
<li><a href="Electric_dipole_moment" title="Electric dipole moment">Electric dipole</a></li>
<li><a href="Electric_field" title="Electric field">Electric field</a></li>
<li><a href="Electric_flux" title="Electric flux">Electric flux</a></li>
<li><a href="Electric_potential" title="Electric potential">Electric potential</a></li>
<li><a href="Electrostatic_discharge" title="Electrostatic discharge">Electrostatic discharge</a></li>
<li><a href="Electrostatic_induction" title="Electrostatic induction">Electrostatic induction</a></li>
<li><a href="Gauss's_law" title="Gauss's law">Gauss's law</a></li>
<li><a href="Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li>
<li><a href="Permittivity" title="Permittivity">Permittivity</a></li>
<li><a href="Polarization_density" title="Polarization density">Polarization</a></li>
<li><a href="Electric_potential_energy" title="Electric potential energy">Potential energy</a></li>
<li><a href="Static_electricity" title="Static electricity">Static electricity</a></li>
<li><a href="Triboelectric_effect" title="Triboelectric effect">Triboelectricity</a></li></ul></div></div></td>
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<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Magnetostatics" title="Magnetostatics">Magnetostatics</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Amp%C3%A8re's_circuital_law" title="Ampère's circuital law">Ampère's law</a></li>
<li><a href="Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot–Savart law</a></li>
<li><a href="Gauss's_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law for magnetism</a></li>
<li><a href="Magnetic_moment" title="Magnetic moment">Magnetic dipole</a></li>
<li><a href="Magnetic_field" title="Magnetic field">Magnetic field</a></li>
<li><a href="Magnetic_flux" title="Magnetic flux">Magnetic flux</a></li>
<li><a href="Magnetic_vector_potential" title="Magnetic vector potential">Magnetic vector potential</a></li>
<li><a href="Magnetization" title="Magnetization">Magnetization</a></li>
<li><a href="Permeability_(electromagnetism)" title="Permeability (electromagnetism)">Permeability</a></li>
<li><a href="Right-hand_rule#Electromagnetism" title="Right-hand rule">Right-hand rule</a></li></ul></div></div></td>
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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Classical_electromagnetism" title="Classical electromagnetism">Electrodynamics</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Bremsstrahlung" title="Bremsstrahlung">Bremsstrahlung</a></li>
<li><a href="Cyclotron_radiation" title="Cyclotron radiation">Cyclotron radiation</a></li>
<li><a href="Displacement_current" title="Displacement current">Displacement current</a></li>
<li><a href="Eddy_current" title="Eddy current">Eddy current</a></li>
<li><a href="Electromagnetic_field" title="Electromagnetic field">Electromagnetic field</a></li>
<li><a href="Electromagnetic_induction" title="Electromagnetic induction">Electromagnetic induction</a></li>
<li><a href="Electromagnetic_pulse" title="Electromagnetic pulse">Electromagnetic pulse</a></li>
<li><a href="Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic radiation</a></li>
<li><a href="Faraday's_law_of_induction" title="Faraday's law of induction">Faraday's law</a></li>
<li><a href="Jefimenko's_equations" title="Jefimenko's equations">Jefimenko equations</a></li>
<li><a href="Larmor_formula" title="Larmor formula">Larmor formula</a></li>
<li><a href="Lenz's_law" title="Lenz's law">Lenz's law</a></li>
<li><a href="Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a></li>
<li><a href="London_equations" title="London equations">London equations</a></li>
<li><a href="Lorentz_force" title="Lorentz force">Lorentz force</a></li>
<li><a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a></li>
<li><a href="Maxwell_stress_tensor" title="Maxwell stress tensor">Maxwell tensor</a></li>
<li><a href="Poynting_vector" title="Poynting vector">Poynting vector</a></li>
<li><a href="Synchrotron_radiation" title="Synchrotron radiation">Synchrotron radiation</a></li></ul></div></div></td>
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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Electrical_network" title="Electrical network">Electrical network</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Alternating_current" title="Alternating current">Alternating current</a></li>
<li><a href="Capacitance" title="Capacitance">Capacitance</a></li>
<li><a href="Current_density" title="Current density">Current density</a></li>
<li><a href="Direct_current" title="Direct current">Direct current</a></li>
<li><a href="Electric_current" title="Electric current">Electric current</a></li>
<li><a href="Electric_power" title="Electric power">Electric power</a></li>
<li><a href="Electrolysis" title="Electrolysis">Electrolysis</a></li>
<li><a href="Electromotive_force" title="Electromotive force">Electromotive force</a></li>
<li><a href="Electrical_impedance" title="Electrical impedance">Impedance</a></li>
<li><a href="Inductance" title="Inductance">Inductance</a></li>
<li><a href="Joule_heating" title="Joule heating">Joule heating</a></li>
<li><a href="Kirchhoff's_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff's laws</a></li>
<li><a href="Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">Network analysis</a></li>
<li><a href="Ohm's_law" title="Ohm's law">Ohm's law</a></li>
<li><a href="Series_and_parallel_circuits#Parallel_circuits" title="Series and parallel circuits">Parallel circuit</a></li>
<li><a href="Electrical_resistance_and_conductance" title="Electrical resistance and conductance">Resistance</a></li>
<li><a href="Resonator#Electromagnetics" title="Resonator">Resonant cavities</a></li>
<li><a href="Series_and_parallel_circuits#Series_circuits" title="Series and parallel circuits">Series circuit</a></li>
<li><a href="Voltage" title="Voltage">Voltage</a></li>
<li><a href="Watt" title="Watt">Watt</a></li>
<li><a href="Waveguide_(radio_frequency)" title="Waveguide (radio frequency)">Waveguides</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content hlist">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Magnetic_circuit" title="Magnetic circuit">Magnetic circuit</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="AC_motor" title="AC motor">AC motor</a></li>
<li><a href="DC_motor" title="DC motor">DC motor</a></li>
<li><a href="Electric_machine" title="Electric machine">Electric machine</a></li>
<li><a href="Electric_motor" title="Electric motor">Electric motor</a></li>
<li><a href="Gyrator%E2%80%93capacitor_model" title="Gyrator–capacitor model">Gyrator–capacitor</a></li>
<li><a href="Induction_motor" title="Induction motor">Induction motor</a></li>
<li><a href="Linear_motor" title="Linear motor">Linear motor</a></li>
<li><a href="Magnetomotive_force" title="Magnetomotive force">Magnetomotive force</a></li>
<li><a href="Permeance" title="Permeance">Permeance</a></li>
<li><a href="Magnetic_complex_reluctance" title="Magnetic complex reluctance">Reluctance (complex)</a></li>
<li><a href="Magnetic_reluctance" title="Magnetic reluctance">Reluctance (real)</a></li>
<li><a href="Rotor_(electric)" title="Rotor (electric)">Rotor</a></li>
<li><a href="Stator" title="Stator">Stator</a></li>
<li><a href="Transformer" title="Transformer">Transformer</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content hlist">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Covariant_formulation_of_classical_electromagnetism" title="Covariant formulation of classical electromagnetism">Covariant formulation</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a></li>
<li><a href="Classical_electromagnetism_and_special_relativity" title="Classical electromagnetism and special relativity">Electromagnetism and special relativity</a></li>
<li><a href="Four-current" title="Four-current">Four-current</a></li>
<li><a href="Electromagnetic_four-potential" title="Electromagnetic four-potential">Four-potential</a></li>
<li><a href="Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions</a></li>
<li><a href="Maxwell's_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell equations in curved spacetime</a></li>
<li><a href="Relativistic_electromagnetism" title="Relativistic electromagnetism">Relativistic electromagnetism</a></li>
<li><a href="Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">Stress–energy tensor</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content hlist">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a></li>
<li><a href="Jean-Baptiste_Biot" title="Jean-Baptiste Biot">Biot</a></li>
<li><a href="Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a></li>
<li><a href="Humphry_Davy" title="Humphry Davy">Davy</a></li>
<li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>
<li><a href="Michael_Faraday" title="Michael Faraday">Faraday</a></li>
<li><a href="Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li>
<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li>
<li><a href="Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a></li>
<li><a href="Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li>
<li><a href="Joseph_Henry" title="Joseph Henry">Henry</a></li>
<li><a href="Heinrich_Hertz" title="Heinrich Hertz">Hertz</a></li>
<li><a href="John_Hopkinson" title="John Hopkinson">Hopkinson</a></li>
<li><a href="Oleg_D._Jefimenko" title="Oleg D. Jefimenko">Jefimenko</a></li>
<li><a href="James_Prescott_Joule" title="James Prescott Joule">Joule</a></li>
<li><a href="Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li>
<li><a href="Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff</a></li>
<li><a href="Joseph_Larmor" title="Joseph Larmor">Larmor</a></li>
<li><a href="Emil_Lenz" title="Emil Lenz">Lenz</a></li>
<li><a href="Alfred-Marie_Li%C3%A9nard" title="Alfred-Marie Liénard">Liénard</a></li>
<li><a href="Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li>
<li><a href="James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li>
<li><a href="Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a></li>
<li><a href="Georg_Ohm" title="Georg Ohm">Ohm</a></li>
<li><a href="Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></li>
<li><a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li>
<li><a href="John_Henry_Poynting" title="John Henry Poynting">Poynting</a></li>
<li><a href="William_Ritchie_(physicist)" title="William Ritchie (physicist)">Ritchie</a></li>
<li><a href="F%C3%A9lix_Savart" title="Félix Savart">Savart</a></li>
<li><a href="George_Singer" title="George Singer">Singer</a></li>
<li><a href="Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a></li>
<li><a href="Nikola_Tesla" title="Nikola Tesla">Tesla</a></li>
<li><a href="J._J._Thomson" title="J. J. Thomson">Thomson</a></li>
<li><a href="Alessandro_Volta" title="Alessandro Volta">Volta</a></li>
<li><a href="Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a></li>
<li><a href="Emil_Wiechert" title="Emil Wiechert">Wiechert</a></li></ul></div></div></td>
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<p><b>Magnetic scalar potential</b>, <i>ψ</i>, is a quantity in <a href="Classical_electromagnetism" title="Classical electromagnetism">classical electromagnetism</a> analogous to <a href="Electric_potential" title="Electric potential">electric potential</a>. It is used to specify the <a href="Magnetic_field#The_H-field" title="Magnetic field">magnetic <b>H</b>-field</a> in cases when there are no <a href="Free_current" class="mw-redirect" title="Free current">free currents</a>, in a manner analogous to using the electric potential to determine the electric field in <a href="Electrostatics" title="Electrostatics">electrostatics</a>. One important use of <i>ψ</i> is to determine the magnetic field due to <a href="Magnet" title="Magnet">permanent magnets</a> when their <a href="Magnetization" title="Magnetization">magnetization</a> is known. The potential is valid in any <a href="Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a> region with zero <a href="Current_density" title="Current density">current density</a>, thus if currents are confined to wires or surfaces, piecemeal solutions can be stitched together to provide a description of the magnetic field at all points in space.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Magnetic_scalar_potential">Magnetic scalar potential</h2></div>
<p>The <a href="Scalar_potential" title="Scalar potential">scalar potential</a> is a useful quantity in describing the magnetic field, especially for <a href="Permanent_magnet" class="mw-redirect" title="Permanent magnet">permanent magnets</a>.
</p><p>Where there is no free current and no <a href="Displacement_current" title="Displacement current">displacement current</a>,
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times \mathbf {H} =\mathbf {0} ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">H</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">0</mn>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {H} =\mathbf {0} ,}</annotation>
</semantics>
</math></span></span>
so if this holds in <a href="Simply_connected_domain" class="mw-redirect" title="Simply connected domain">simply connected domain</a> we can define a <em>magnetic scalar potential</em>, <span class="texhtml"><i>ψ</i></span>, as<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} =-\nabla \psi .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">H</mi>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>ψ<!-- ψ --></mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} =-\nabla \psi .}</annotation>
</semantics>
</math></span></span>
The dimension of <span class="texhtml"><i>ψ</i></span> in <a href="SI_base_units" class="mw-redirect" title="SI base units">SI base units</a> is <span class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {A}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="sans-serif">A</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathsf {A}}}</annotation>
</semantics>
</math></span><img src="./102043e55e1bdbf81aad8c9c1419ba91d44d6755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle {\mathsf {A}}}" loading="lazy"></span>,</span> which can be expressed in SI units as <a href="Ampere" title="Ampere">amperes</a>.
</p><p>Using the definition of <span class="texhtml"><b>H</b></span>:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {B} =\mu _{0}\nabla \cdot \left(\mathbf {H} +\mathbf {M} \right)=0,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mo>=</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">H</mi>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">M</mi>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {B} =\mu _{0}\nabla \cdot \left(\mathbf {H} +\mathbf {M} \right)=0,}</annotation>
</semantics>
</math></span></span>
it follows that
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}\psi =-\nabla \cdot \mathbf {H} =\nabla \cdot \mathbf {M} .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>ψ<!-- ψ --></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">H</mi>
</mrow>
<mo>=</mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">M</mi>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\psi =-\nabla \cdot \mathbf {H} =\nabla \cdot \mathbf {M} .}</annotation>
</semantics>
</math></span></span>
</p><p>Here, <span class="texhtml">∇ ⋅ <b>M</b></span> acts as the source for magnetic field, much like <span class="texhtml">∇ ⋅ <b>P</b></span> acts as the source for electric field. So analogously to <a href="Bound_charge" class="mw-redirect" title="Bound charge">bound electric charge</a>, the quantity
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{m}=-\nabla \cdot \mathbf {M} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">M</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho _{m}=-\nabla \cdot \mathbf {M} }</annotation>
</semantics>
</math></span></span>
is called the <i>bound magnetic charge</i> density. Magnetic charges <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle q_{m}=\int \rho _{m}\,dV}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>∫<!-- ∫ --></mo>
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle q_{m}=\int \rho _{m}\,dV}</annotation>
</semantics>
</math></span><img src="./6caa148e695977029e1255d5a24e4994f0856585.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.883ex; height:3.176ex;" alt="{\textstyle q_{m}=\int \rho _{m}\,dV}" loading="lazy"></span> never occur isolated as <a href="Magnetic_monopole" title="Magnetic monopole">magnetic monopoles</a>, but only within dipoles and in magnets with a total magnetic charge sum of zero. The energy of a localized magnetic charge <span class="texhtml"><i>q<sub>m</sub></i></span> in a magnetic scalar potential is
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\mu _{0}\,q_{m}\psi ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo>=</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mi>ψ<!-- ψ --></mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q=\mu _{0}\,q_{m}\psi ,}</annotation>
</semantics>
</math></span></span>
and of a magnetic charge density distribution <span class="texhtml"><i>ρ<sub>m</sub></i></span> in space
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\mu _{0}\int \rho _{m}\psi \,dV,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo>=</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>∫<!-- ∫ --></mo>
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mi>ψ<!-- ψ --></mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>V</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q=\mu _{0}\int \rho _{m}\psi \,dV,}</annotation>
</semantics>
</math></span></span>
where <span class="texhtml"><i>µ</i><sub>0</sub></span> is the <a href="Vacuum_permeability" title="Vacuum permeability">vacuum permeability</a>. This is analog to the energy <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=qV_{E}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo>=</mo>
<mi>q</mi>
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>E</mi>
</mrow>
</msub>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q=qV_{E}}</annotation>
</semantics>
</math></span><img src="./e0e92ace122b8d4a2d5d524a6c9ed578c96903da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.849ex; height:2.509ex;" alt="{\displaystyle Q=qV_{E}}" loading="lazy"></span> of an electric charge <span class="texhtml"><i>q</i></span> in an electric potential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{E}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>E</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V_{E}}</annotation>
</semantics>
</math></span><img src="./b327692ccc05af0bf8f8fdf45b159d0e6ed849ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.843ex; height:2.509ex;" alt="{\displaystyle V_{E}}" loading="lazy"></span>.
</p><p>If there is free current, one may subtract the contributions of free current per <a href="Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot–Savart law</a> from total magnetic field and solve the remainder with the scalar potential method.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Magnetic_vector_potential" title="Magnetic vector potential">Magnetic vector potential</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFVanderlinde2005">Vanderlinde 2005</a>, pp. 194–199</span>
</li>
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">
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/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFDuffin1980" class="citation book cs1">Duffin, W.J. (1980). <i>Electricity and Magnetism, Fourth Edition</i>. McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>007084111X</bdi>.</cite></li></ul>
<ul><li><cite id="CITEREFJackson1999" class="citation cs2">Jackson, John David (1999), <i>Classical Electrodynamics</i> (3rd ed.), <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-30932-X</bdi></cite></li></ul>
<ul><li><cite id="CITEREFVanderlinde2005" class="citation book cs1">Vanderlinde, Jack (2005). <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/1250088"><i>Classical Electromagnetic Theory</i></a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005cet..book.....V">2005cet..book.....V</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F1-4020-2700-1">10.1007/1-4020-2700-1</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-4020-2699-4</bdi>.</cite></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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