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Illustration Second Maxwell equation in integral form

Second Maxwell equation in integral form
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Maxwell equation for the divergence of the magnetic field \( \nabla \cdot \boldsymbol{B} \). This Maxwell equation states that there are no magnetic monopoles. Magnetic fields always occur as dipoles with a south pole S and a north pole N. Thus the divergence of the magnetic field or the flux integral is always zero:\[ \nabla \cdot \boldsymbol{B} ~=~ 0 \]or equivalently: \[ \oint_{A} \boldsymbol{B} ~\cdot~ \text{d}\boldsymbol{a} ~=~ 0 \]

The scalar product \(\boldsymbol{B} ~\cdot~ \text{d}\boldsymbol{a}\) only picks out the component \(\boldsymbol{B}_{||}\) of the magnetic field \(\boldsymbol{B}\) that is parallel to the \(\text{d}\boldsymbol{a}\) element.