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gumthakka
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Well, I guess that is pretty much my question.
Of course this is not correct. The full Maxwell equation readsrude man said:Ampere's law always holds BUT if your wire is not of infinite length then there has to be a circuit to complete the current loop. Those extra conductors will contribute to the B field the same way the wire does that you're integrating around, spoiling the symmetry. But ## \oint \bf H \cdot \bf dl = I ## always.
We should not forget that the "unbroken" symmetry in the infinite wire is only an approximation motivated by one's desire to apply Ampere's law and get the answer ##B=\dfrac{\mu_0I}{2\pi r}.~## A current requires a closed loop, else there will be violation of charge conservation. For a loop where ##a## is a linear parameter determining the size of the loop, the approximation is valid close to the wire where ##r<<a.##Delta2 said:The symmetry is broken when the wire is of finite length. Due to broken symmetry, the magnetic field will not be the same along the closed amperian loop on which we perform integration, so ampere's law integral can't be simplified to ##B\cdot 2\pi r=\mu_0 I##. But it always hold that ##\oint _{\partial S} \mathbf{B}\cdot \mathbf{dl}=\mu_0\left (I+ \epsilon_0\int\int_S \frac{\partial \mathbf{E}}{\partial t}\cdot\hat n dS\right )## for any closed surface ##S## with boundary the loop ##\partial S##.
It is just that $$\oint \mathbf{B}\cdot\mathbf{dl}\neq \mathbf{B}\cdot 2\pi r$$ when the (cylindrical) symmetry is broken.
Ampere's circuital law is a fundamental law in electromagnetism that describes the relationship between the magnetic field and the electric current in a closed loop. It states that the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the permeability of free space.
Ampere's circuital law may fail to hold in certain cases, such as when there is a time-varying electric field or when the medium surrounding the current-carrying conductor is not homogeneous or isotropic.
An example of when Ampere's circuital law fails to hold is when there is a changing electric field, such as in the case of a capacitor being charged or discharged. In this case, the changing electric field creates a displacement current, which contributes to the magnetic field and violates the assumption of a steady current in Ampere's circuital law.
In cases where Ampere's circuital law fails to hold, it is modified to include the displacement current term. This modified version is known as Maxwell's circuital law and is one of the four Maxwell's equations that describe the behavior of electromagnetic fields.
Understanding when Ampere's circuital law fails to hold is important because it allows us to accurately model and predict the behavior of electromagnetic fields in more complex situations. It also led to the development of Maxwell's equations, which have been crucial in advancing our understanding and application of electromagnetism in various fields such as telecommunications, electronics, and power generation.