Just as the electric flux on a Gaussian surface is calculated, an alternative statement of the Biot-Savart law could summarize the magnetic field via a closed directed path or a so-called amperic loop. And similarly, instead of the total load contained in a Gaussian surface, we examine the total current trapped by an amperic loop. As with a Gaussian surface, an amperian loop has an orientation towards the loop. In general, the positive alignment of the buckle is defined as the counterclockwise direction. This convention can be remembered by fighting the fingers of the right hand around the thumb (the so-called “rule of law”). There are a number of ambiguities in the above definitions that require clarification and choice of convention. While it`s not mathematically strict, I hope it`s enough. Note that we are only dealing with differential forms, not integral forms, but this is enough because the differential and integral forms are each equivalent according to the Kelvin–Stokes theorem. In free space, the displacement current is related to the time rate of change of the electric field. Proof that the formulations of the law of the circuit with respect to the free current correspond to the formulations with total current Note that care must be taken to correctly align the loop with respect to the direction of the current. If the loop is counterclockwise on the side, all currents coming out on the side are positive, while those coming in on the side are negative. where J mathbf{J} J is the current density.
This is the generalized and continuous version of the current I I I. Of course, the surface integral in both equations can be adopted on any selected closed surface, so the integrands must be the same: in the first loop, the path itself is quite small, but the magnetic field is strong near the wire. In the large loop below, the path is much larger, but the magnetic field is much weaker. In fact, according to the Biot-Savart law, we know that the field attenuates by a factor of exactly 1/r1/r 1/r, where are is the radius of the loop. Thus, the linear change in the size of a circular loop is precisely compensated by the decrease in the strength of the magnetic field at the point of the loop. In other words, the sum of the magnetic field on the direction of a circular loop is James Clerk Maxwell designed the displacement current as a polarization current in the sea dielectric vortex, with which he modeled the magnetic field hydrodynamically and mechanically. [17] He added this displacement current to Ampère`s circuit law to equation 112 in his 1861 work “On Physical Lines of Force.” [18] The integral form of the original circuit law is a linear integral of the magnetic field around a closed curve C (arbitrary, but must be closed). The C curve, in turn, limits both an area S, through which the electric current flows (again arbitrarily, but not closed – since no three-dimensional volume is surrounded by S), and the current. The mathematical statement of the law is a relationship between the total amount of the magnetic field around a path (line integral) due to the current flowing through that closed path (surface integral). [10] [11] (integral form), where H is the magnetic field H (also called “auxiliary magnetic field”, “magnetic field strength” or simply “magnetic field”), D is the electric displacement field and Jf is the closed line current or free current density. In differential form, the first term on the right side is present everywhere, even in a vacuum. It does not involve any real movement of the charge, but it still has an associated magnetic field as if it were a real current.
Some authors currently apply the name change only to this article. [19] In cgs units, the integral form of the equation is, including Maxwell`s correction. where the current density JD is the displacement current and J is the current density contribution actually due to the movement of the charges, both free and bound. Since ∇ ⋅ D = ρ, the problem of charge continuity with Ampère`s original formulation is no longer a problem. [22] Due to the term in ε0∂E/∂t, the propagation of waves in free space is now possible. ∇×B=μ0J+μ0ε0∂E∂t. nabla times mathbf{B} = mu_0 mathbf{J} + mu_0 epsilon_0 frac{partial mathbf{E}}{partial t}. ∇×B=μ0J+μ0ε0∂t∂E. In the wire (r≤R), (r leq R), (r≤R) the total current included I(r2/R2) is Ibig(r^2/R^2big) I(r2/R2), so the law of ampere gives the density of the polarization current. Let`s take the equation for B: with Jf the “free” or “conductive” current density. The meaning of Maxwell`s additional term lies in the fact that it allows you to do the following.
For simplicity, suppose there is an area of space where the electric field E(x) E(x) E(x) E(x) is non-zero only along the z z-z axis, and the magnetic field B(x) B(x) B(x) B(x) B(x) is non-zero only along the y axis, so both are only functions of x x x. Next, Faraday`s law gives 2πrμ0I2πr=μ0I, 2 pi r frac{mu_0 I}{2 pi r} = mu_0 I, 2πr2πrμ0I=μ0I, Maxwell added a term called displacement current to represent the contribution of the time-varying electric field to ∇×B nabla times mathbf{B} ∇×B. The size of the additional term, he argued, should be ε0∂E/∂t epsilon_0 partial mathbf{E}/partial t ε0∂E/∂t. In this case, Ampere`s law is that when a material is magnetized (for example, by placing it in an external magnetic field), the electrons remain bound to their respective atoms, but behave as if orbiting the nucleus in a certain direction and generating a microscopic current.