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The electrons in the current in a wire are constantly colliding with the metal cations which means there must be a constant cause of acceleration: an electric field. How, why is it that the electric field is identical in shape to that of the wire?

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The electric field has to be provided by a battery for example. There is no electric field within a conductor by itself. If a voltage is applied to a bent conductor, surface charges accumulate in curved parts which leads to a net electric field in axial direction.

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The electric field $\vec{E}$ also exists outside the wire. There is however no current there as there are no charges. Inside the wire, for simplicity assumed to be an isotropic conductor, $\vec{E}$ is related to the current density $\vec{j}$ by $\vec{j} = \sigma \vec{E}$, where $\sigma$ is the conductivity. It is not zero unless $\sigma$ is infinite as is the case for a superconductor.

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