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I am given the following question:

Two long coaxial cylindrical conductors of radii $a$ and $b$ ($b$ < $a$) are thin and hollow. There is a vacuum between the conductors ($b$ < $r$ < $a$, where r is the radial distance from the axis). The conductors are maintained at a potential difference $V$$0$ and carry equal but opposing currents of magnitude $I$ (in the direction of the axis of the cylinders). Find the magnitude and direction of the electric and magnetic fields in the region between the conductors, defining carefully the direction of the currents and the sign of the potential difference you have assumed.

From what I know, there's a magnetic field present between the two cylinders, which is produced by the current passing through the inner cylinder (all I need to do is to use Ampere's law). Current in the outer cylinder doesn't affect the magnetic field between them. However, how can there be an electric field present?

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There's none... It's infinitely long... no charge can accumulate. So "magnitude" of electric field=0.

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Clearly, magnetic field occurs only between the two cylinders just like as a current carrying wire, and the electic field occurs also in the region due to potential difference $V_0$. For a steady state current the electric field only exists along the direction of current.

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There is an electric field because there is a non-zero PD between the cylinders. There can be an accumulation of charge on the surface of the conductors as well as a movement of charge (current) inside the conductors. These are not incompatible.

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