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since we can get the charge conservation from ampere's law i was wondering is this the only physical interpretation of ampere's law ,what can be the more physical interpretations of ampere's law both in differential and integral form

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  • $\begingroup$ What do you mean by a "physical interpretation"? What specifically do you want us to explain? $\endgroup$ Dec 31, 2016 at 11:31
  • $\begingroup$ I want the meaning of equation in both forms the best way i can think of , the way i can visualise each term and the relation between the entities described. $\endgroup$
    – Syed Ilyas
    Dec 31, 2016 at 12:06

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It is convention not to reference corollaries of an equation (like charge conservation) as " a physical interpretation" of the equation. In differential form, Ampere's law states that

$$\nabla\times\mathbf{B}=\mu_0\mathbf{J} +\mu_0\epsilon_0\frac{\partial\mathbf{E}}{\partial t}$$

Personally, I think that what this equation states physically is not obvious. For this reason, when people look for physical interpretations of one of the Maxwell equations, they refer to the integral form. In integral form, Ampere's law is

$$\oint_{\partial R}\mathbf{B}\cdot d\mathbf{l}=\mu_0\iint_R\mathbf{J}\cdot d\mathbf{a}\ + \mu_0\epsilon_0\frac{\partial}{\partial t}\iint_R\mathbf{E}\cdot d\mathbf{a}$$

Note that $R$ denote's to the region of integration, and $\partial R$ is just notation that refers to the boundary of the region of integration $R$. What the second equation states is a bit more transparent. Now give your self some region $R$, and orient the curve enclosing that area $\partial R$ in some direction (say clockwise for definiteness).

This equation states that if you go around the curve $\partial R$ and measure how much the magnetic field is in the direction of an arrow pointing in the "forward" direction (i.e. the direction of orientation) and then add all of that up, that is the same quantity just counting how much current is passing through $R$ $plus$ how many electric field lines are passing through $R$ per unit time.

Most succinctly, one could simply say that magnetic fields are caused by currents and time changing electric fields through some region $R$. This is perhaps best for what you should carry in your head and should remember as the qualitative content of Ampere's law.

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