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Tangential component of $\mathbf{B}$ is discontinuous at the boundary of two media. Does this mean that tangential component of $\mathbf{B}$ is undefined at the boundary of two media?

If yes, then:

$\mathbf{B}$ is undefined at the boundary of two media.

$\nabla \cdot\mathbf{B}$ is undefined at the boundary of two media.

This contradicts $\nabla \cdot\mathbf{B}=0$ everywhere. How to get out of this difficulty?

If no, then:

What is the value of tangential component of $\mathbf{B}$ at the boundary of two media?

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    $\begingroup$ Are you saying that tangential component of $\mathbf{B}$ is not discontinuous? $\endgroup$ – N.G.Tyson Apr 28 at 6:54
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    $\begingroup$ No, that might be wrong. I deleted my comment. $\endgroup$ – Hardey Pandya Apr 28 at 7:00
  • $\begingroup$ Just because $\mathbf B$ is discontinuous at a point it is not necessarily true that $\mathbf B$ has no value there. In physics, electric and magnetic field can have value even at a point where it is discontinous. For example, electric field on surface of a charged conducting sphere or magnetic field on surface of a current carrying cylinder (ideal solenoid). These value are determined based on force acting on the surface, which can be determined. $\endgroup$ – Ján Lalinský Apr 28 at 12:39
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Does this mean that tangential component of 𝐁 is undefined at the boundary of two media?

An infinitely sharp boundary with a delta function surface current? Yes.

This contradicts ∇⋅𝐁=0 everywhere. How to get out of this difficulty?

What's the difficulty? Maxwell's equations are differential equations for vector fields, and hold on a particular domain. In the model you're describing, the sharp boundary isn't actually in the domain of $\mathbf B$.


If you're uncomfortable with this, there are workarounds. If you define the tangential component of $\mathbf B$ on the boundary to be the average of the tangential components on either side, and interpret the derivatives in Maxwell's equations in a weak sense, then everything is well-defined.

Alternatively, because nothing in nature is ever infinitely sharp, you could replace the sharp boundaries and surface currents with things like bump functions, which would eliminate discontinuities by smoothing everything out.

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    $\begingroup$ It seems to me that the divergence of B is defined at the sharp boundary. And it's zero. $\endgroup$ – Rob Jeffries Apr 28 at 10:02
  • $\begingroup$ Also, in some cases $\mathbf B$ can be defined at a boundary of two media, based on the magnetic force acting on the surface current there. For example, $\mathbf B$ is defined on the surface of current-carrying cylinder (perfect solenoid). $\endgroup$ – Ján Lalinský Apr 28 at 12:37
  • $\begingroup$ @RobJeffries If the tangential component of the B-field is undefined, then I don't see how you can differentiate it. What do you plug into the difference quotient? I agree that if you adopt the averaging procedure, or define the boundary value of $\mathbf B$ to be equal to either the right or left hand limit, then everything is fine. $\endgroup$ – J. Murray Apr 28 at 15:33
  • $\begingroup$ @JánLalinský I haven't thought about that before, so I'll have to consider it further. $\endgroup$ – J. Murray Apr 28 at 15:34
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No the tangential component of B is well defined. It is just discontinuous. As it is in the direction perpendicular to the tangential component, the discontinuity does not affect $\vec \nabla \cdot \vec B$, as it happens in the direction perpendicular to $\vec B$.

Note that in this case many quantities are discontinuous: current, atomic density, electron density, dielectric response, to name a few. Of course the discontinuity is a limit of your model of the media. At atomic scale there is no discontinuity.

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    $\begingroup$ (1) If it is discontinuous the left and right hand limits will be having different values. Then what will be the value at the point of discontinuity? (2) Why this discontinuity does not affect $\vec{\nabla} \cdot \vec{B}$? $\endgroup$ – N.G.Tyson Apr 28 at 7:17
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    $\begingroup$ I extended my answer to address your comment. $\endgroup$ – my2cts Apr 28 at 7:28
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    $\begingroup$ You said "Of course the discontinuity is a limit of your model of the media. At atomic scale there is no discontinuity." I am dealing only with classical physics in which the distributions are considered perfectly continuous. So my question is: "In classical electromagnetism" how shall one find the value of tangential component of $\mathbf{B}$ at the point of discontinuity? $\endgroup$ – N.G.Tyson Apr 28 at 7:41
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    $\begingroup$ Seems as if your first para of the answer is incomplete. Please consider rechecking it. $\endgroup$ – N.G.Tyson Apr 28 at 7:52
  • $\begingroup$ > "I am dealing only with classical physics in which the distributions are considered perfectly continuous." But that is an artificial restriction of classical physics. There are plenty examples where discontinuous functions are used in classical physics, there is nothing wrong or non-classical with that. $\endgroup$ – Ján Lalinský Apr 28 at 12:34
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The interface condition for the tangential component of the magnetic field is derived from the Ampere-Maxwell equation. Here is a detailed derivation on the same.

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A discontinuity in the tangential component of the magnetic field intensity at the boundary must be supported by surface current flowing in a direction perpendicular to that component of the field. For an infinitesimally thin interface between two macroscopic regions (labelled 1 and 2) carrying as surface charge density $K$, denoting the unit normal from region 1 and 2 as $\hat{n}_1$ and $\hat{n}_2$ respectively, it is usually stated as $$ \hat{n}_2\times [\vec{B_1} - \vec{B_2}] = \mu \vec{K}$$

Suppose there is no surface current. Then the tangential component of the magnetic field intensity is continuous across the boundary unless the permeabilities are unequal. Also, note that the particular expression for discontinuity works for only to an interface which is at rest in the frame of reference where the fields are measured.

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The tangential boundary condition for ${\bf B}$ can be obtained from the tangential boundary condition for ${\bf H}$ which is a consequence of Ampère's law, by taking a small rectangular circuit with two sides parallel to the surface: $$ {\bf n \times( H_2 - H_1 )= J_s}~~~~~~~~~~~~~[1] $$ where $J_s$ is the surface current density which may be present on the surface separating region 1 and region 2. The unit vector {\bf n} is normal to the sepaation surface and points from region 1 to region 2. This relation for ${\bf H}$ can be rewritten in term of ${\bf B}$ by using the relation between ${\bf H}$ and ${\bf B}$ fields in the two regions. For example, for linear media where ${\bf B}= \mu {\bf H}$, eq.[1] becomes $$ {\bf n} \times( { \bf B_2}/\mu_2 - {\bf B_1}/\mu_1 {\bf )= J_s}~~~~~~~~~~~~~[2] $$

As you can see, the (possible) jump of ${\bf B}$ is determined by this condition.

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The tangential component of the B-field is undefined at an infinitely sharp interface, but the divergence of the B-field is still zero.

Consider a simple example. An interface between two media with different permeabilities. The interface is defined by the plane $z=0$. The B-field either side of the interface is wholly parallel to the plane of the interface. Let us also say there are no moving charges anywhere.

Maxwell's equations (Ampere's law) can be used to show that $\vec{H}_1= \vec{H}_2$, where in both cases, the H-fields have components along the $x$ and $y$ axes.

If we are dealing with linear, isotropic, homogeneous media then $\vec{B} =\mu \vec{H}$ and so $\mu_2 \vec{B}_1 = \mu_1 \vec{B}_2$ and the B-field changes, as a step function, as we move from one medium to another.

This step function is an artefact of the model for the interface. If the interface is modelled as a step function we should not be surprised to find step functions in other quantities.

However that does not mean that $\nabla \cdot \vec{B}$ is undefined, since it must satisfy the "solenoidal law" of Maxwell's equations.

The B-field only changes discontinuously along the $z$-axis, and since $B_z = 0$, then $\partial B_z/\partial z =0$ and the divergence of the B-field is still zero everywhere, as it must be.

The introduction of a z-component to the field either side of the interface will make no difference, because we say that this component is continuous by demanding that the divergence of the B-field is zero at the interface!

Ditto for adding surface currents. They only change the B-fields along the x and y axes and in a divergence-free manner.

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  • $\begingroup$ Magnetic field $\mathbf B$ can be defined even at the interface, based on force that would act on current-carrying conductor there. If we put in between the two media a very thin sheet of metal and run small enough current through it, this will not change the old magnetic field in any measurable way, but we will be able to measure magnetic force acting on this metal sheet due to old field, and then use this to complete the definition of $\mathbf B$ on the boundary surface. $\endgroup$ – Ján Lalinský Apr 28 at 12:45
  • $\begingroup$ @JánLalinský The question specifies that the B-field is discontinuous. By making the boundary less-than-infinitely narrow then don't you make the B-field continuous and/or create two new discontinuities at the interfaces with the metal. What force does the tangential magnetic field exert on a current-carrying sheet that it is parallel to? $\endgroup$ – Rob Jeffries Apr 28 at 13:06
  • $\begingroup$ In theory we can put in infinitely thin current sheet at the boundary where surface current of $\boldsymbol{\lambda}$ amperes per meter flows. Magnetic force on unit surface due to external field is then $\boldsymbol{\lambda} \times\mathbf B$. Since force should be unique, magnetic field should be unique as well. $\endgroup$ – Ján Lalinský Apr 28 at 13:21

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