Ampere`s Law Curl

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”). This is called the differential form of Ampere`s law, because it expresses Ampere`s law in the form of a differential operator, the loop. PS: This is called here the macroscopic circulation without loop: This is the differential form of Ampere`s law and one of Maxwell`s equations. It indicates that the curvature of the magnetic field at any point is the same as the current density there. Another way to affirm this law is that current density is a source of ripple of the magnetic field.

Your paddle wheel should be infinitely small. You could put a similar paddle wheel in some charge distributions for the electric field and rotate them if we know that the curvature of the electrostatic field must always be $0. The problem is that the loop is a respectful operator that works with infinitesimal quantities. Therefore, you need to think in terms of an infinitesimal paddle wheel. Then you will see that you are not rotating at points that are not on the z-axis. On the right side of equality in equation [4], we used Stokes` theorem to change a straight-line integral around a closed loop in the curvature of the same field through the surface surrounded by the loop (S). I understand that Ampère`s law tells us that the current density multiplied by $mu_0$ at a place equal to the loop of $mathbf{B}$ at that place. Conceptually, however, it worries me. Let`s calculate the curvature of this field. In cylindrical coordinates, the curvature of a vector field with a single component $phi$ is given by $$nablatimesmathbf B=-frac{partial B_phi}{partial z}has rho+frac{1}{rho}left(frac{partial}{partial rho}(rho B_phi)right)has z$$ Since $B_phi=frac{mu_0I}{2pirho}$, we have $$frac{partial B_phi}{partial z}=0$$ $$frac{partial}{partial rho}(rho B_phi)=0$$ Therefore, we have $nablatimesmathbf B=0$ at all points of the coin, except where $rho=0$, where the $1/rho$ part causes us some problems. However, you can do some tricky calculations to show that the curvature on the z-axis results in a Dirac delta distribution, so as expected, we end up with $$nablatimesmathbf B=frac{frac{mu_0I}{2pirho}delta(rho)has z=mu_0mathbf J$$.

So everything is going well. Ampère`s law was written up to Maxwell as in equation [6]. So let`s see what`s wrong with that. First of all, I must reject another vector identity – the divergence of the curvature of any vector field is always zero: in one of the great triumphs of physics, perhaps the brightest between the epochs of Newton and Einstein, in the mid-nineteenth century, Maxwell showed that the “naïve” form of Ampère`s law that was used is necessarily inadequate. or at least not completely generalized. It turns out that, analogous to Faraday`s law, in which a time-varying magnetic field leads to an electric field, a time-varying electric field also leads to a magnetic field. In other words, a positive curvature in the magnetic field can be the result not only of stationary currents, but also of time-varying electric fields. Now we have a new form of Ampere`s law: the curvature of the magnetic field is equal to the density of electric current. If you are a shrewd learner, you may find that equation [6] is not the final form written in equation [1]. There is a problem with equation [6], but it was not until the 1860s that James Clerk Maxwell understood the problem and unified electromagnetism with Maxwell`s equations. Stokes` theorem tells us that the circulation integral can be replaced by the integral of the curvature of the vector field on any surface bounded by the loop.

We will choose the surface so that it corresponds to the surface used to compress the flow of the flow. One of Maxwell`s equations, Ampere`s law, relates the curvature of the magnetic field to current density and is particularly useful for current distributions with high degrees of symmetry. In particular, I imagine a long thread in my head through which a certain current density passes $J $. Then, outside of this thread, $$B should be proportional to $s^{-1}$, where $s$ is the distance of the wire (the integral of $mathbf{B}$ on each loop around the wire should be $mu_0$ once I included). At all points outside the line, $J$ is zero, so $text{curl}(mathbf{B})$ must also be zero. Maxwell knew that the electric field (and the electric flux density (D) in the capacitor changed. And he knew that a time-varying magnetic field led to a magnetic electric field (that is, this is Farday`s law – the curvature of E corresponds to the time derivative of B). So why is it not true that a time-varying D field would lead to a magnetic H field (i.e. the formation of the curvature of H). The universe loves symmetry, so why not introduce this term? And that`s what Maxwell did, and he called this term displacement current density: so there SHOULD be a loop, or at least I`d say without mathematics.

But since mathematics is not wrong, where is my reasoning wrong? To express Ampere`s law in a local form, one can use Stokes` theorem to describe the right integral ∫B⋅ds int mathbf{B} cdot dmathbf{s} ∫B⋅ds with respect to the surface integral of B`s curvature mathbf{B} B: There are two important questions regarding the law of circuits that require further study. First, there is a problem with the continuity equation for the electric charge. In vector calculus, the identity of the divergence of a curvature indicates that the divergence of the curvature of a vector field must always be zero. Therefore, $text{curl} = $0$ at any given time implies that placing a paddle wheel at this point does not cause the paddle wheel to rotate. However, consider a cross-section of the wire with the $$B field that revolves around it.