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I know this question is asked a lot, but I am confused as to why ampere's law cannot be applied to certain situations. In particular: why can't it be used to calculate the field from a wire of finite length?
nrqed said:Ampere's law is valid (well, if you include the term added by Maxwell), no matter what. *BUT* the integral over \vec B \cdot \dl is in general difficult to do. It is only in a simple case like an (idealized) infinite wire that you can say that (with a circular closed loop centered around the wire) that vec B \cdot \dl = B dl and that, moreover, the magnitude of the B field is a constant which may be taken out of the integral.
It's the same as for Gauss' law for the electric field. It is *always* valid but the integral is easy to do only in specific cases with a lot of symmetry (spherical, cylindrical or planar symmetry). When a system is more complex and there is no obviosu symmetry, it is not that the law fails, it is rather than it is not very useful because it involves an integral very difficult to do. The reason books look at those special cases (infinte planes, infinite wires, infinite cylinders, etc) is that these are the only cases where the integrals involved in Gauss' and Ampere's laws are easy to carry out. It does not mean that the laws are not valid al the time, simply that they are not terribly useful.
Ampere's Law is a fundamental law of electromagnetism that describes the relationship between current and magnetic fields. However, it is only applicable to situations where the magnetic field is constant and the current is steady. In situations where the magnetic field is varying or the current is changing, Ampere's Law cannot accurately predict the behavior of the magnetic field.
No, Ampere's Law is only applicable to situations with steady currents. When the current is changing, it produces a changing magnetic field, which violates the assumptions of Ampere's Law. In these situations, Maxwell's equations must be used to accurately describe the behavior of the magnetic field.
Ampere's Law cannot be applied to situations with changing magnetic fields, such as those produced by varying currents or time-varying electric fields. It also cannot be used to analyze the behavior of magnetic materials, as these materials can affect the behavior of the magnetic field.
Ampere's Law is based on a set of simplifying assumptions that make it easier to apply in certain situations. However, these assumptions also limit its applicability to more complex situations. In order to accurately describe the behavior of the magnetic field in more complex situations, other laws and equations, such as Maxwell's equations, must be used.
There are modifications of Ampere's Law, such as Ampere-Maxwell Law, which take into account the effects of changing electric fields on the behavior of the magnetic field. However, even with these modifications, Ampere's Law is still limited in its applicability and other laws and equations must be used in more complex situations.