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You have a coil over an iron hearth. There is a current in coil which creates the flux $\phi_{1}$. The flux then distributes over the wider area in the iron (using wrong word?): $\phi_{2}$ the flux over the middle and the $\phi_{3}$ over the left.

            |-----------------|
            |        |        | 
\phi_{3}    |\phi_{2}|        | COIL HERE 
            |        |        |      the lines are of the iron hearth
            |-----------------|

                           \phi_{1}

The flux does not disappear so

$$\phi_{1} = \phi_{2} + \phi_{3}$$

could some explain the last statement? The last statement is a bit like Kirhoff's law. But I am unsure how can you play with fluxes. Could someone elaborate on this? On which rule, is the statement based on?

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A fundamental postulate of electromagnetism is that the flux of the magnetic field through any closed surface is zero. This is essentially a statement that there are no magnetic monopoles. This implies that the divergence of the magnetic field is zero. The corresponding equation is

$$\nabla \cdot \vec{B} = 0$$

This is not very much like Kirchoff's (Voltage) Law, which is based on the vanishing curl of the electric field

$$\nabla \times \vec{E} = 0$$

and is only true in the absence of a time-dependent magnetic field.

To summarize the difference: The law about magnetic fields is saying that you take a closed plastic bag, contort it into any shape, and however much magnetic flux enters the bag has to flow back out again somewhere else. The law about electric fields is saying that you take loop of wire, contort it into any shape, and however much the electric field pushes a charged particle forward in one part of the loop, it will push the particle backwards by the same amount over the rest of the loop.

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