Magnetic potential – wikipedia gas exchange in the lungs takes place in the


where B is the magnetic field and E is the electric field. In magnetostatics where there is no time-varying charge distribution, only the 3 gases that cause acid rain first equation is needed. (In the context of electrodynamics, the terms vector potential and scalar potential are used for magnetic vector potential and electric potential, respectively. In mathematics, vector potential and scalar potential can be generalized to higher dimensions.)

If electric and magnetic fields are defined as above from potentials gas station near me, they automatically satisfy two of Maxwell’s equations: Gauss’s law for magnetism and Faraday’s Law. For example, if A is continuous and well-defined everywhere, then it is guaranteed not to result in magnetic monopoles. (In the mathematical theory of magnetic monopoles, A is allowed to be either undefined or multiple-valued in some places; see magnetic monopole for details).

∇ ⋅ B = ∇ ⋅ ( ∇ × A ) = 0 ∇ × E = ∇ × ( − ∇ ϕ − ∂ A ∂ t ) = − ∂ ∂ t ( ∇ × A ) = − ∂ B ∂ t . {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {B} =\nabla \cdot \left(\nabla \times \mathbf {A} \right)=0\\\nabla \times \mathbf {E} =\nabla \times \left(-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}\right)=-{\frac {\partial }{\partial t}}\left(\nabla \times \mathbf {A} \right)=-{\frac {\partial \mathbf {B} }{\partial t}}.\end{aligned}}}

Although the magnetic field B is a pseudovector (also called axial vector), the vector electricity estimated bills potential A is a polar vector. [3] This means that if the right-hand rule for cross products were replaced with a left-hand rule, but without changing any other electricity office equations or definitions, then B would switch signs, but A would not change. This is an example of a general theorem: The curl of a polar vector is a pseudovector, and vice versa. [3] Gauge choices [ edit ]

The solutions of Maxwell’s equations in the Lorenz gauge (see Feynman [2] and Jackson [4]) with the boundary condition that both potentials go to zero sufficiently fast as they approach infinity are called the retarded potentials, which are the magnetic vector potential A( r, t) and the electric scalar potential ϕ( r, t) due to a current distribution of current density electricity physics pdf J( r′, t′), charge density ρ( r′, t′), and volume Ω, within which ρ and J are non-zero at least sometimes and some places):

A ( r , t ) = μ 0 4 π ∫ Ω J ( r ′ , t ′ ) | r − r ′ | d 3 r ′ ϕ ( r , t ) = 1 4 π ϵ 0 ∫ Ω ρ ( r ′ , t ′ ) | r − r ′ | d 3 r ′ {\displaystyle {\begin{aligned}\mathbf {A} \left(\mathbf {r} ,t\right)={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {\mathbf {J} \left(\mathbf {r} ‘,t’\right)}{\left|\mathbf {r} -\mathbf {r} ‘\right|}}\,\mathrm {d} ^{3}\mathbf {r} ‘\\\phi \left(\mathbf {r} ,t\right)={\frac {1}{4\pi \epsilon _{0}}}\int _{\Omega }{\frac {\rho \left(\mathbf {r} ‘,t’\right)}{\left|\mathbf {r} -\mathbf {r} ‘\right|}}\,\mathrm {d} ^{3}\mathbf {r} ‘\end{aligned}}}

• The equation for A is a vector equation. In Cartesian coordinates, the equation separates into three electricity omd scalar equations: [5] A x ( r , t ) = μ 0 4 π ∫ Ω J x ( r ′ , t ′ ) | r − r ′ | d 3 r ′ A y ( r , t ) = μ 0 4 π ∫ Ω J y ( r ′ , t ′ ) | r − r ′ | d 3 r ′ A z ( r , t ) = μ 0 4 π ∫ Ω J z ( r ′ , t ′ ) | r − r ′ | d 3 r ′ {\displaystyle {\begin{aligned}A_{x}\left(\mathbf {r} ,t\right)={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {J_{x}\left(\mathbf {r} ‘,t’\right)}{\left|\mathbf {r} -\mathbf {r} ‘\right|}}\,\mathrm {d} ^{3}\mathbf {r} ‘\\A_{y}\left(\mathbf {r} ,t\right)={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {J_{y}\left(\mathbf o gosh {r} ‘,t’\right)}{\left|\mathbf {r} -\mathbf {r} ‘\right|}}\,\mathrm {d} ^{3}\mathbf {r} ‘\\A_{z}\left(\mathbf {r} ,t\right)={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {J_{z}\left(\mathbf {r} ‘,t’\right)}{\left|\mathbf {r} -\mathbf {r} ‘\right|}}\,\mathrm {d} ^{3}\mathbf {r} ‘\end{aligned}}}