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Hi all,
I'm working through chapter 2 of Michael Tinkham's Introduction to Superconductivity. On page 40, he asserts that the skin-depth for a general complex conductivity is (In Gaussian units)
$$\delta = \frac{c}{\sqrt{2\pi\omega\left(|\sigma| + \sigma_2\right)}}$$
where $$\sigma = \sigma_1 - i\sigma_2$$
I am trying to derive this skin-depth expression, but can't seem to get it. My process is as follows. I have bolded the two places where potential issues could be. Start with combining Faraday's law and Ampere's law:
$$\vec{\nabla}\times\vec{E} = -\frac{1}{c}\frac{\partial\vec{B}}{\partial t}$$
$$ \vec{\nabla}\times\vec{\nabla}\times\vec{E} = -\frac{1}{c}\frac{\partial}{\partial t}\left(\frac{4\pi}{c}\vec{J} + \frac{1}{c}\frac{\partial\vec{E}}{\partial t}\right)$$
Now use J = \sigma E and assume E = exp(i\omega t),
$$ \vec{\nabla}(\vec{\nabla}\cdot\vec{E}) - \nabla^2\vec{E} = -\frac{1}{c}\left(\frac{i4\pi\sigma\omega}{c}\vec{E} - \frac{\omega^2}{c}\vec{E}\right) $$
Now a potentially sketchy step, I assume the displacement current term is very small (I'm dealing with a superconductor here, so it makes sense), and I also assume the charge density is spatially uniform. This results in eliminating the first and fourth terms. I think this may be where I'm losing it, but I'm not sure how else to do it. The result is
$$\nabla^2\vec{E} = \frac{i4\pi\sigma\omega}{c^2}\vec{E}$$
Solving this,
$$\vec{E} = \vec{E}_0\exp{\left(-\sqrt{\frac{i4\pi\sigma\omega}{c^2}}z\right)} $$
$$= \vec{E}_0 \exp{\left(-\sqrt{\frac{4\pi\omega(\sigma_2 + i\sigma_1)}{c^2}}z\right)}$$
I'm having trouble now. I try to separate this into real and imaginary parts, but the real part does not seem to be what Tinkham has. I think the way I'm taking the squareroot of a complex number is the problem.
$$= \vec{E}_0 \exp{\left(-\sqrt{\frac{4\pi\omega|\sigma|}{c^2}}\left(\cos{\theta/2} + i\sin{\theta/2}\right)\right)}$$
Where $$\theta = \arctan{\frac{\sigma_1}{\sigma_2}}$$
The resulting skin depth is
$$\delta = \frac{c}{\sqrt{4\pi\omega |\sigma |}}\frac{1}{\cos{\theta/2}}$$
This makes sense to me, is close to the given value, and reduces to the skin depth of a real conductivity for \sigma_2 = 0, as it should. My expression does reduce to Tinkham's if I assume \sigma_1 << \sigma_2, which is a very reasonable approximation. But the assertion in the book that this is a "genera" solution is what troubles me.
Does anyone have ideas on what I'm missing? I would really appreciate your help!
I'm working through chapter 2 of Michael Tinkham's Introduction to Superconductivity. On page 40, he asserts that the skin-depth for a general complex conductivity is (In Gaussian units)
$$\delta = \frac{c}{\sqrt{2\pi\omega\left(|\sigma| + \sigma_2\right)}}$$
where $$\sigma = \sigma_1 - i\sigma_2$$
I am trying to derive this skin-depth expression, but can't seem to get it. My process is as follows. I have bolded the two places where potential issues could be. Start with combining Faraday's law and Ampere's law:
$$\vec{\nabla}\times\vec{E} = -\frac{1}{c}\frac{\partial\vec{B}}{\partial t}$$
$$ \vec{\nabla}\times\vec{\nabla}\times\vec{E} = -\frac{1}{c}\frac{\partial}{\partial t}\left(\frac{4\pi}{c}\vec{J} + \frac{1}{c}\frac{\partial\vec{E}}{\partial t}\right)$$
Now use J = \sigma E and assume E = exp(i\omega t),
$$ \vec{\nabla}(\vec{\nabla}\cdot\vec{E}) - \nabla^2\vec{E} = -\frac{1}{c}\left(\frac{i4\pi\sigma\omega}{c}\vec{E} - \frac{\omega^2}{c}\vec{E}\right) $$
Now a potentially sketchy step, I assume the displacement current term is very small (I'm dealing with a superconductor here, so it makes sense), and I also assume the charge density is spatially uniform. This results in eliminating the first and fourth terms. I think this may be where I'm losing it, but I'm not sure how else to do it. The result is
$$\nabla^2\vec{E} = \frac{i4\pi\sigma\omega}{c^2}\vec{E}$$
Solving this,
$$\vec{E} = \vec{E}_0\exp{\left(-\sqrt{\frac{i4\pi\sigma\omega}{c^2}}z\right)} $$
$$= \vec{E}_0 \exp{\left(-\sqrt{\frac{4\pi\omega(\sigma_2 + i\sigma_1)}{c^2}}z\right)}$$
I'm having trouble now. I try to separate this into real and imaginary parts, but the real part does not seem to be what Tinkham has. I think the way I'm taking the squareroot of a complex number is the problem.
$$= \vec{E}_0 \exp{\left(-\sqrt{\frac{4\pi\omega|\sigma|}{c^2}}\left(\cos{\theta/2} + i\sin{\theta/2}\right)\right)}$$
Where $$\theta = \arctan{\frac{\sigma_1}{\sigma_2}}$$
The resulting skin depth is
$$\delta = \frac{c}{\sqrt{4\pi\omega |\sigma |}}\frac{1}{\cos{\theta/2}}$$
This makes sense to me, is close to the given value, and reduces to the skin depth of a real conductivity for \sigma_2 = 0, as it should. My expression does reduce to Tinkham's if I assume \sigma_1 << \sigma_2, which is a very reasonable approximation. But the assertion in the book that this is a "genera" solution is what troubles me.
Does anyone have ideas on what I'm missing? I would really appreciate your help!