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Qmechanic
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Current charge density of the H$H$-field

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SuperCiocia
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In this Wikipedia article: https://en.wikipedia.org/wiki/Magnetic_field#:~:text=south%20to%20north.-,H%2Dfield%20and%20magnetic%20materials,-Comparison%20of%20B

we see how the volume free current charge density and the surface free current charge density are found. While I am able to understand the derivation of the volume free current charge density, I cannocannot understand how was the surface free current charge density was derived:

$$\vec H^{||}_1 - H^{||}_2=\vec K_f \times \vec n$$.$$\vec H^{||}_1 - H^{||}_2=\vec K_f \times \vec n.$$

I have to questions regarding this equation:

  1. How was the part in the right side derived?
  2. How can I understand (geometrically) how is that cross product happening? I don't know how the surface free current charge density $\vec K_f$ flows in the surface to even understand to do the cross product and also know in which direction does the resulting vector points at?

In this Wikipedia article: https://en.wikipedia.org/wiki/Magnetic_field#:~:text=south%20to%20north.-,H%2Dfield%20and%20magnetic%20materials,-Comparison%20of%20B

we see how the volume free current charge density and the surface free current charge density are found. While I am able to understand the derivation of the volume free current charge density, I canno understand how was the surface free current charge density was derived:

$$\vec H^{||}_1 - H^{||}_2=\vec K_f \times \vec n$$.

I have to questions regarding this equation:

  1. How was the part in the right side derived?
  2. How can I understand (geometrically) how is that cross product happening? I don't know how the surface free current charge density $\vec K_f$ flows in the surface to even understand to do the cross product and also know in which direction does the resulting vector points at?

In this Wikipedia article: https://en.wikipedia.org/wiki/Magnetic_field#:~:text=south%20to%20north.-,H%2Dfield%20and%20magnetic%20materials,-Comparison%20of%20B

we see how the volume free current charge density and the surface free current charge density are found. While I am able to understand the derivation of the volume free current charge density, I cannot understand how was the surface free current charge density was derived:

$$\vec H^{||}_1 - H^{||}_2=\vec K_f \times \vec n.$$

I have to questions regarding this equation:

  1. How was the part in the right side derived?
  2. How can I understand (geometrically) how is that cross product happening? I don't know how the surface free current charge density $\vec K_f$ flows in the surface to even understand to do the cross product and also know in which direction does the resulting vector points at?
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imbAF
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Current charge density of the H-field

In this Wikipedia article: https://en.wikipedia.org/wiki/Magnetic_field#:~:text=south%20to%20north.-,H%2Dfield%20and%20magnetic%20materials,-Comparison%20of%20B

we see how the volume free current charge density and the surface free current charge density are found. While I am able to understand the derivation of the volume free current charge density, I canno understand how was the surface free current charge density was derived:

$$\vec H^{||}_1 - H^{||}_2=\vec K_f \times \vec n$$.

I have to questions regarding this equation:

  1. How was the part in the right side derived?
  2. How can I understand (geometrically) how is that cross product happening? I don't know how the surface free current charge density $\vec K_f$ flows in the surface to even understand to do the cross product and also know in which direction does the resulting vector points at?