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From Griffiths' Intro to electrodynamics:

enter image description here

Now I'm confused about 3 things:

1) What is the 'mobile' surface charge density? Isn't the surface current density itself the 'mobile' surface charge density?

2) Is the small current $dI$ in the whole ribbon or only part of the ribbon? I'm guessing its the whole ribbon of width $dl$ but I want to make sure.

3) If I'm asked to find the surface current density as follows:

enter image description here

Is the current flowing only on the surface of the wire or is it flowing naturally through the whole volume of the wire and I need to only account for the current on the surface? , I'm able to derive $K$ will be equal to $I/2\pi a$ mathematically, but if the current is flowing throughout the volume of the wire, how can I visualize the unit length perpendicular to that flow? Thanks I hope I was clear about the questions

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1) What is the 'mobile' surface charge density? Isn't the surface current density itself the 'mobile' surface charge density?

Not all charges are mobile. For instance, protons and most electrons in a solid metal are not mobile. So, here, surface charge density could be interpreted as a surface density of free electrons.

2) Is the small current dI in the whole ribbon or only part of the ribbon? I'm guessing its the whole ribbon of width dl but I want to make sure.

Yes, dl is the width of the whole ribbon.

3) ...Is the current flowing only on the surface of the wire or is it flowing naturally through the whole volume of the wire and I need to only account for the current on the surface?

The current density is not always uniformly distributed through the whole volume of a conductor: most of a high frequency AC current, due to the skin effect, flows in a thin layer under the surface of a conductor. In such cases, it makes sense to talk about a surface current or a surface current density.

I'm able to derive K will be equal to I/2πa mathematically, but if the current is flowing throughout the volume of the wire, how can I visualize the unit length perpendicular to that flow?

As mentioned earlier, the current can flow mostly along the surface of a wire, in which case, the conventional current density, a current through a unit area of the wire's cross-section, could be replaced by the surface current density, a current through a unit length of the wire's circumference.

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