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I was trying to solve a problem in which I had to obtain the function of the magnetic flux through a loop over time. The way the magnetic field and the surface are defined is piece-wise, and I wondered whether the magnetic flux function I wanted to get had to be continuous, since it is defined as an integral (and therefore Fundamental Theorem of Calculus would imply magnetic flux's continuity):

$$\mathbf { \Phi(t) = \int_S B(t)dS}$$ The Magnetic Flux $\Phi$ is the Sum ( Average) of the $\mathbf{B}$- field over the area $\mathbf{S}$

If flux were to be continuous, then checking the continuity of the function I got as a result would be a nice way to know if I've solved the problem correctly or not. That's why I'm asking.

In summary, must magnetic flux over time always be a continuous function? Thanks in advance.

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The magnetic flux through a given area can change with time (as in a transformer). Only the total magnetic flux around a magnetic loop needs to be continuous (field lines form closed loops). The flux through the primary coil of a transformer is ideally equal to that through the secondary coil. (There may be some field lines which pass outside of the secondary, and in an AC situation there may be some loss to radiation.)

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must magnetic flux over time always be a continuous function?

If a magnetic field were to be discontinuous, when considered as a function of time, then the electric field associated with it would be infinite or undefined for a moment.

Such a result cannot dictate what can happen in reality. Reality may defy our expectations. However, it is a strong argument for making the assumption that magnetic fields (and consequently flux), vary only continuously with respect to time.

(That is, when analysing some specific problem, one can say, "I am here assuming that the magnetic field is continuous, because if it were not, the electric field would be undefined. Thus...")

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