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<title>Potential gradient</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Potential gradient</span></span>

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<p>In <a href="Physics" title="Physics">physics</a>, <a href="Chemistry" title="Chemistry">chemistry</a> and <a href="Biology" title="Biology">biology</a>, a <b>potential gradient</b> is the local <a href="Rate_(mathematics)" title="Rate (mathematics)">rate of change</a> of the <a href="Potential" title="Potential">potential</a> with respect to <a href="Displacement_(geometry)" title="Displacement (geometry)">displacement</a>, i.e. spatial <a href="Derivative" title="Derivative">derivative</a>, or <a href="Gradient" title="Gradient">gradient</a>. This quantity frequently occurs in equations of physical processes because it leads to some form of <a href="Flux" title="Flux">flux</a>.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<div class="mw-heading mw-heading3"><h3 id="One_dimension">One dimension</h3></div>

<p>The simplest definition for a potential gradient <i>F</i> in one dimension is the following:<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>

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<annotation encoding="application/x-tex">{\displaystyle F={\frac {\phi _{2}-\phi _{1}}{x_{2}-x_{1}}}={\frac {\Delta \phi }{\Delta x}}\,\!}</annotation>

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</math></span><img src="./1690024bdfca831628927e1e26365f46a51a3370.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-right: -0.387ex; width:21.038ex; height:5.843ex;" alt="{\displaystyle F={\frac {\phi _{2}-\phi _{1}}{x_{2}-x_{1}}}={\frac {\Delta \phi }{\Delta x}}\,\!}" loading="lazy"></span></dd></dl>

<p>where <span class="texhtml"><i>ϕ</i>(<i>x</i>)</span> is some type of <a href="Scalar_potential" title="Scalar potential">scalar potential</a> and <span class="texhtml"><i>x</i></span> is <a href="Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> (not <a href="Distance" title="Distance">distance</a>) in the <span class="texhtml"><i>x</i></span> direction, the subscripts label two different positions <span class="texhtml"><i>x</i><sub>1</sub>, <i>x</i><sub>2</sub></span>, and potentials at those points, <span class="texhtml"><i>ϕ</i><sub>1</sub> = <i>ϕ</i>(<i>x</i><sub>1</sub>), <i>ϕ</i><sub>2</sub> = <i>ϕ</i>(<i>x</i><sub>2</sub>)</span>. In the limit of <a href="Infinitesimal" title="Infinitesimal">infinitesimal</a> displacements, the ratio of differences becomes a ratio of <a href="Differential_of_a_function" title="Differential of a function">differentials</a>:

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<annotation encoding="application/x-tex">{\displaystyle F={\frac {{\rm {d}}\phi }{{\rm {d}}x}}.\,\!}</annotation>

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</math></span><img src="./556cc556776a72320a9784e6edd4bb5c68e82cd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:9.387ex; height:5.509ex;" alt="{\displaystyle F={\frac {{\rm {d}}\phi }{{\rm {d}}x}}.\,\!}" loading="lazy"></span></dd></dl>

<p>The direction of the electric potential gradient is from <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 x_{1}}">

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</math></span><img src="./a8788bf85d532fa88d1fb25eff6ae382a601c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}" loading="lazy"></span> to <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 x_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Three_dimensions">Three dimensions</h3></div>

<p>In <a href="Three_dimensional_space" class="mw-redirect" title="Three dimensional space">three dimensions</a>, <a href="Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> make it clear that the resultant potential gradient is the sum of the potential gradients in each direction:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\mathbf {e} _{x}{\frac {\partial \phi }{\partial x}}+\mathbf {e} _{y}{\frac {\partial \phi }{\partial y}}+\mathbf {e} _{z}{\frac {\partial \phi }{\partial z}}\,\!}</annotation>

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</math></span><img src="./9b6b46ece10c0f1cb0e4ae870259dcd37ceae868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-right: -0.387ex; width:28.367ex; height:6.176ex;" alt="{\displaystyle \mathbf {F} =\mathbf {e} _{x}{\frac {\partial \phi }{\partial x}}+\mathbf {e} _{y}{\frac {\partial \phi }{\partial y}}+\mathbf {e} _{z}{\frac {\partial \phi }{\partial z}}\,\!}" loading="lazy"></span></dd></dl>

<p>where <span class="texhtml"><b>e</b><sub>x</sub>, <b>e</b><sub>y</sub>, <b>e</b><sub>z</sub></span> are <a href="Unit_vector" title="Unit vector">unit vectors</a> in the <span class="texhtml"><i>x, y, z</i></span> directions. This can be compactly written in terms of the <a href="Gradient" title="Gradient">gradient</a> <a href="Operator_(mathematics)" title="Operator (mathematics)">operator</a> <span class="texhtml">∇</span>,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\nabla \phi .\,\!}</annotation>

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</math></span><img src="./ff29393ff97f3466109880bac11aa02625c4f651.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:9.136ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =\nabla \phi .\,\!}" loading="lazy"></span></dd></dl>

<p>although this final form holds in any <a href="Curvilinear_coordinate_system" class="mw-redirect" title="Curvilinear coordinate system">curvilinear coordinate system</a>, not just Cartesian.

</p><p>This expression represents a significant feature of any <a href="Conservative_vector_field" title="Conservative vector field">conservative vector field</a> <span class="texhtml"><b>F</b></span>, namely <span class="texhtml"><b>F</b></span> has a corresponding potential <span class="texhtml"><i>ϕ</i></span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Using <a href="Stokes'_theorem" title="Stokes' theorem">Stokes' theorem</a>, this is equivalently stated as

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {F} ={\boldsymbol {0}}\,\!}</annotation>

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</math></span><img src="./f57e568c8f01015f24253f26b607b275188f550c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:11.281ex; height:2.176ex;" alt="{\displaystyle \nabla \times \mathbf {F} ={\boldsymbol {0}}\,\!}" loading="lazy"></span></dd></dl>

<p>meaning the <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a>, denoted ∇×, of the vector field vanishes.

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<div class="mw-heading mw-heading2"><h2 id="Physics">Physics</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Newtonian_gravitation">Newtonian gravitation</h3></div>

<p>In the case of the <a href="Gravitational_field#classical_mechanics" title="Gravitational field">gravitational field</a> <span class="texhtml"><b>g</b></span>, which can be shown to be conservative,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> it is equal to the gradient in <a href="Gravitational_potential" title="Gravitational potential">gravitational potential</a> <span class="texhtml">Φ</span>:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} =-\nabla \Phi .\,\!}</annotation>

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</math></span><img src="./67226619afdb0b3cb30f2b7acc078b444f4100f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:10.891ex; height:2.509ex;" alt="{\displaystyle \mathbf {g} =-\nabla \Phi .\,\!}" loading="lazy"></span></dd></dl>

<p>There are opposite signs between gravitational field and potential, because the potential gradient and field are opposite in direction: as the potential increases, the gravitational field strength decreases and vice versa.

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<div class="mw-heading mw-heading3"><h3 id="Electromagnetism">Electromagnetism</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a> and <a href="Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions of the electromagnetic field</a></div>

<p>In <a href="Electrostatics" title="Electrostatics">electrostatics</a>, the <a href="Electric_field" title="Electric field">electric field</a> <span class="texhtml"><b>E</b></span> is independent of time <span class="texhtml"><i>t</i></span>, so there is no induction of a time-dependent <a href="Magnetic_field" title="Magnetic field">magnetic field</a> <span class="texhtml"><b>B</b></span> by <a href="Faraday's_law_of_induction" title="Faraday's law of induction">Faraday's law of induction</a>:

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}={\boldsymbol {0}}\,,}</annotation>

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</math></span><img src="./b2d09ac1f12ba59b65525762ce87c9786e484d87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.964ex; height:5.509ex;" alt="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}={\boldsymbol {0}}\,,}" loading="lazy"></span></dd></dl>

<p>which implies <span class="texhtml"><b>E</b></span> is the gradient of the electric potential <span class="texhtml"><i>V</i></span>, identical to the classical gravitational field:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle -\mathbf {E} =\nabla V.\,\!}</annotation>

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<p>In <a href="Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>, the <span class="texhtml"><b>E</b></span> field is time dependent and induces a time-dependent <span class="texhtml"><b>B</b></span> field also (again by Faraday's law), so the curl of <span class="texhtml"><b>E</b></span> is not zero like before, which implies the electric field is no longer the gradient of electric potential. A time-dependent term must be added:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle -\mathbf {E} =\nabla V+{\frac {\partial \mathbf {A} }{\partial t}}\,\!}</annotation>

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</math></span><img src="./5349b1634469c4f0ddff1d5250ae2c1757af87de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:17.788ex; height:5.509ex;" alt="{\displaystyle -\mathbf {E} =\nabla V+{\frac {\partial \mathbf {A} }{\partial t}}\,\!}" loading="lazy"></span></dd></dl>

<p>where <span class="texhtml"><b>A</b></span> is the electromagnetic <a href="Vector_potential" title="Vector potential">vector potential</a>. This last potential expression in fact reduces Faraday's law to an identity.

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<div class="mw-heading mw-heading3"><h3 id="Fluid_mechanics">Fluid mechanics</h3></div>

<p>In <a href="Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>, the <a href="Velocity_field" class="mw-redirect" title="Velocity field">velocity field</a> <span class="texhtml"><b>v</b></span> describes the fluid motion. An <a href="Irrotational_flow" class="mw-redirect" title="Irrotational flow">irrotational flow</a> means the velocity field is conservative, or equivalently the <a href="Vorticity" title="Vorticity">vorticity</a> <a href="Pseudovector" title="Pseudovector">pseudovector</a> field <span class="texhtml"><b>ω</b></span> is zero:

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {v} ={\boldsymbol {0}}.}</annotation>

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</math></span><img src="./4d79ab20b83e560abb382d3452c891c6d1c9b980.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.036ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {v} ={\boldsymbol {0}}.}" loading="lazy"></span></dd></dl>

<p>This allows the <a href="Velocity_potential" title="Velocity potential">velocity potential</a> to be defined simply as:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =\nabla \phi }</annotation>

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</math></span><img src="./e09d9800f5aff5981465cb7525684146c7c405e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.831ex; height:2.509ex;" alt="{\displaystyle \mathbf {v} =\nabla \phi }" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Chemistry">Chemistry</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Electrode_potentials" class="mw-redirect" title="Electrode potentials">Electrode potentials</a></div>

<p>In an <a href="Electrochemistry" title="Electrochemistry">electrochemical</a> <a href="Half-cell" title="Half-cell">half-cell</a>, at the interface between the <a href="Electrolyte" title="Electrolyte">electrolyte</a> (an <a href="Ion" title="Ion">ionic</a> <a href="Solution_(chemistry)" title="Solution (chemistry)">solution</a>) and the <a href="Metal" title="Metal">metal</a> <a href="Electrode" title="Electrode">electrode</a>, the standard <a href="Electric_potential_difference" class="mw-redirect" title="Electric potential difference">electric potential difference</a> is:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle \Delta \phi _{(M,M^{+z})}=\Delta \phi _{(M,M^{+z})}^{\ominus }+{\frac {RT}{zeN_{\text{A}}}}\ln a_{M^{+z}}\,\!}</annotation>

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</math></span><img src="./491f0f7fa03286f3719d6db475b04d603e88359e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-right: -0.387ex; width:41.656ex; height:5.676ex;" alt="{\displaystyle \Delta \phi _{(M,M^{+z})}=\Delta \phi _{(M,M^{+z})}^{\ominus }+{\frac {RT}{zeN_{\text{A}}}}\ln a_{M^{+z}}\,\!}" loading="lazy"></span></dd></dl>

<p>where <i>R</i> = <a href="Gas_constant" title="Gas constant">gas constant</a>, <i>T</i> = <a href="Temperature" title="Temperature">temperature</a> of solution, <i>z</i> = <a href="Valence_(chemistry)" title="Valence (chemistry)">valency</a> of the metal, <i>e</i> = <a href="Elementary_charge" title="Elementary charge">elementary charge</a>, <i>N</i><sub>A</sub> = <a href="Avogadro_constant" title="Avogadro constant">Avogadro constant</a>, and <i>a</i><sub>M<sup>+z</sup></sub> is the <a href="Activity_(chemistry)" class="mw-redirect" title="Activity (chemistry)">activity</a> of the ions in solution. Quantities with superscript ⊖ denote the measurement is taken under <a href="Standard_temperature_and_pressure" title="Standard temperature and pressure">standard conditions</a>. The potential gradient is relatively abrupt, since there is an almost definite boundary between the metal and solution, hence the interface term.

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<div class="mw-heading mw-heading2"><h2 id="Biology">Biology</h2></div>

<p>In <a href="Biology" title="Biology">biology</a>, a potential gradient is the net difference in <a href="Electric_charge" title="Electric charge">electric charge</a> across a <a href="Cell_membrane" title="Cell membrane">cell membrane</a>.

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<div class="mw-heading mw-heading2"><h2 id="Non-uniqueness_of_potentials">Non-uniqueness of potentials</h2></div>

<p>Since gradients in potentials correspond to <a href="Field_(physics)" title="Field (physics)">physical fields</a>, it makes no difference if a constant is added on (it is erased by the gradient operator <span class="texhtml">∇</span> which includes <a href="Partial_differentiation" class="mw-redirect" title="Partial differentiation">partial differentiation</a>). This means there is no way to tell what the "absolute value" of the potential "is" – the zero value of potential is completely arbitrary and can be chosen anywhere by convenience (even "at infinity"). This idea also applies to vector potentials, and is exploited in <a href="Classical_field_theory" title="Classical field theory">classical field theory</a> and also <a href="Gauge_field_theory" class="mw-redirect" title="Gauge field theory">gauge field theory</a>.

</p><p>Absolute values of potentials are not physically observable, only gradients and path-dependent potential differences are. However, the <a href="Aharonov%E2%80%93Bohm_effect" title="Aharonov–Bohm effect">Aharonov–Bohm effect</a> is a <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> effect which illustrates that non-zero <a href="Electromagnetic_potential" class="mw-redirect" title="Electromagnetic potential">electromagnetic potentials</a> along a closed loop (even when the <span class="texhtml"><b>E</b></span> and <span class="texhtml"><b>B</b></span> fields are zero everywhere in the region) lead to changes in the phase of the <a href="Wave_function" title="Wave function">wave function</a> of an electrically <a href="Charged_particle" title="Charged particle">charged particle</a> in the region, so the potentials appear to have measurable significance.

</p>

<div class="mw-heading mw-heading2"><h2 id="Potential_theory">Potential theory</h2></div>

<p><a href="Field_equation" title="Field equation">Field equations</a>, such as Gauss's laws <a href="Gauss's_law" title="Gauss's law">for electricity</a>, <a href="Gauss's_law_for_magnetism" title="Gauss's law for magnetism">for magnetism</a>, and <a href="Gauss's_law_for_gravity" title="Gauss's law for gravity">for gravity</a>, can be written in the form:

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<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {F} =X\rho }</annotation>

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</math></span><img src="./b3106e2db308a1905da39a018f6aa3b05bd74a36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.578ex; height:2.676ex;" alt="{\displaystyle \nabla \cdot \mathbf {F} =X\rho }" loading="lazy"></span></dd></dl>

<p>where <span class="texhtml"><i>ρ</i></span> is the electric <a href="Charge_density" title="Charge density">charge density</a>, <a href="Magnetic_monopole" title="Magnetic monopole">monopole</a> density (should they exist), or <a href="Mass_density" class="mw-redirect" title="Mass density">mass density</a> and <span class="texhtml"><i>X</i></span> is a constant (in terms of <a href="Physical_constant" title="Physical constant">physical constants</a> <span class="texhtml"><a href="Gravitational_constant" title="Gravitational constant"><i>G</i></a></span>, <span class="texhtml"><a href="Vacuum_permittivity" title="Vacuum permittivity"><i>ε</i><sub>0</sub></a></span>, <span class="texhtml"><a href="Vacuum_permeability" title="Vacuum permeability"><i>μ</i><sub>0</sub></a></span> and other numerical factors).

</p><p>Scalar potential gradients lead to <a href="Poisson's_equation" title="Poisson's equation">Poisson's equation</a>:

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<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot (\nabla \phi )=X\rho \quad \Rightarrow \quad \nabla ^{2}\phi =X\rho }</annotation>

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</math></span><img src="./141b01aae1c853dcd3c60725da110591207a530b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.941ex; height:3.176ex;" alt="{\displaystyle \nabla \cdot (\nabla \phi )=X\rho \quad \Rightarrow \quad \nabla ^{2}\phi =X\rho }" loading="lazy"></span></dd></dl>

<p>A general <a href="Potential_theory" title="Potential theory">theory of potentials</a> has been developed to solve this equation for the potential. The gradient of that solution gives the physical field, solving the field equation.

</p>

<ul><li><a href="Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</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">Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, <style data-mw-deduplicate="TemplateStyles:r1238218222">

/* start https://en.wikipedia.org/ */

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7195-3382-1</bdi></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipcshutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-161545-7</bdi></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-01460-8</bdi></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-92712-9</bdi></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>81-7758-293-3</bdi></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Physical chemistry, P.W. Atkins, Oxford University Press, 1978, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-855148-7</bdi></span>

</li>

</ol></div></div></div><div><div class="zim-footer">

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