Register to reply

Ampere's Law with Maxwell's correction is equivelant to Ampere's Law?

by Question Man
Tags: ampere's law, biot savart law
Share this thread:
Question Man
Feb28-12, 11:30 AM
P: 9
Is it true that Ampere's Law with Maxwell's correction is equivelant to Biot-Savart Law?
Under what assumptions?
Phys.Org News Partner Physics news on
New complex oxides could advance memory devices
Nature's designs inspire research into new light-based technologies
UCI team is first to capture motion of single molecule in real time
Feb28-12, 03:20 PM
Sci Advisor
P: 2,546
Biot-Savart holds for stationary fields, where Maxwell's displacement current doesn't play a role, i.e., you have the two magnetostatic equations (here for simplicity I neglect medium effects, i.e., use the vacuum equations in Heaviside-Lorentz units)

[tex]\vec{\nabla} \times \vec{B}=\frac{\vec{j}}{c}, \quad \vec{\nabla} \cdot \vec{B}=0.[/tex]

From the second equation, which says that there are no magnetic charges, we see that the magnetic field is a pure solenoidal field, i.e., there is a vector potential, [itex]\vec{A}[/itex] such that

[tex]\vec{B}=\vec{\nabla} \times \vec{A}.[/tex]

For a given magnetic field, the vector potential is only determined up to the gradient of a scalar field, and thus we can choose a constraint on the potential. In the so called Coulomb gauge one assumes

[tex]\vec{\nabla} \cdot \vec{A}=0.[/tex]

Plugging now this ansatz into the first equation, which is Ampere's Law, we get (in Cartesian coordinates!)

[tex]\vec{\nabla} \times (\vec{\nabla} \times \vec{A})=\vec{\nabla} (\vec{\nabla}
\cdot \vec{A})-\Delta \vec{A}=-\Delta \vec{A}=\frac{\vec{j}}{c}.[/tex]

Now this looks like the equation of electrostatics for each Cartesian component of the vector potential. From this we get immediately the solution in terms of the Green's function of the Laplacian:

[tex]\vec{A}(\vec{x})=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \frac{\vec{j}(\vec{x}')}{4 \pi c |\vec{x}-\vec{x}'|}.[/tex]

Taking the curl of this solution, directly yields the Biot-Savart Law,

[tex]\vec{B}(\vec{x})=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \vec{j}(\vec{x}') \times \frac{\vec{x}-\vec{x}'}{4 \pi c |\vec{x}-\vec{x}'|^3} .[/tex]
Question Man
Feb29-12, 12:24 PM
P: 9
Muchas Gracias!

Register to reply

Related Discussions
Magnetism and Ampere-Maxwell's law Astronomy & Astrophysics 7
Ampere-Maxwell Law Classical Physics 2
Ampere-Maxwell law Advanced Physics Homework 5
Ampere/Maxwell law...capacitor problem Introductory Physics Homework 2
Application of Ampère-Maxwell equ. Classical Physics 3