Electromagnetic four-potential

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The electromagnetic four-potential is a four-vector defined in SI units (and gaussian units in parentheses) as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): A^{\alpha}=\left(\frac{\phi}{c}, \vec A \right) \qquad \left(A^a=(\phi, \vec A)\right)

in which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \phi is the electrical potential, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \vec A is the magnetic potential, a vector potential.

The electric and magnetic fields associated with these four-potentials are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \vec{E} = -\vec{\nabla} \phi - \frac{\partial \vec{A}}{\partial t} \qquad \left( -\vec{\nabla} \phi - \frac{1}{c} \frac{\partial \vec{A}}{\partial t} \right)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \vec{B} = \vec{\nabla} \times \vec{A}

It is useful to group the potentials together in this form because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): A_{\alpha} is a Lorentz covariant vector, meaning that it transforms in the same way as the spacetime coordinates (t, x) under transformations in the Lorentz group: rotations and Lorentz boosts. As a result, the inner product

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): A^{\alpha} A_{\alpha} = |\vec{A}|^2 -\frac{\phi^2}{c^2} \qquad \left(A^a A_a \, = |\vec{A}|^2 - \phi^2 \right)

is the same in every inertial reference frame.

Often, physicists employ the Lorenz gauge condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \partial_{\alpha} A^{\alpha} = 0 to simplify Maxwell's equations as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \Box^2 A_{\alpha} = -\mu_0 J_{\alpha} \qquad \left( \Box^2 A_{\alpha} = -\frac{4 \pi}{c} J_{\alpha} \right)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): J_\alpha are the components of the four-current,

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \Box^2 = \nabla^2 -\frac{1}{c^2} \frac{\partial^2} {\partial t^2} is the d'Alembertian operator.

In terms of the scalar and vector potentials, this last equation becomes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \Box^2 \phi = -\frac{\rho}{\epsilon_0} \qquad \left(\Box^2 \phi = -4 \pi \rho \right)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \Box^2 \vec{A} = -\mu_0 \vec{j} \qquad \left( \Box^2 \vec{A} = -\frac{4 \pi}{c} \vec{j} \right)

For a given charge and current distribution, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \rho(\vec{x},t) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \vec{j}(\vec{x},t) , the solutions to these equations in SI units are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \phi (\vec{x}, t) = \frac{1}{4 \pi \epsilon_0} \int \mathrm{d}^3 x^\prime \frac{\rho( \vec{x}^\prime, \tau)}{ \left| \vec{x} - \vec{x}^\prime \right|}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \vec A (\vec{x}, t) = \frac{\mu_0}{4 \pi} \int \mathrm{d}^3 x^\prime \frac{\vec{j}( \vec{x}^\prime, \tau)}{ \left| \vec{x} - \vec{x}^\prime \right|} ,

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \tau = t - \frac{\left|\vec{x}-\vec{x}'\right|}{c} is the retarded time. This is sometimes also expressed with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \rho(\vec{x}',\tau)=[\rho(\vec{x}',t)] , where the square brackets are meant to indicate that the time should be evaluated at the retarded time. Of course, since the above equations are simply the solution to an inhomogeneous differential equation, any solution to the homogeneous equation can be added to these to satisfy the boundary conditions. These homogeneous solutions in general represent waves propagating from sources outside the boundary.

When the integrals above are evaluated for typical cases, e.g. of an oscillating current (or charge), they are found to give both a magnetic field component varying as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): r^{-2} (the induction field) and a component decreasing as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): r^{-1} (the radiation field).

References

  • Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press. ISBN 0-19-853952-5. 
  • Jackson, J D (1999). Classical Electrodynamics (3rd). New York: Wiley. ISBN ISBN 0-471-30932-X. 

See also

fa:چاربردار پتانسیل


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