Electric field in a cylindrical conductor

[Nikitin]

Homework Statement

Problem 1c from here: http://web.phys.ntnu.no/~ingves/Teaching/TFY4240/Exam/Exam_tfy4240_Dec_2013.pdf

Homework Equations

Maxwell's equations

The Attempt at a Solution

According to the solutions, the electric field is ZERO everywhere because it's "magnetostatics" and because the net charge is zero everywhere. http://web.phys.ntnu.no/~ingves/Teaching/TFY4240/Exam/Solution_tfy4240_Dec_2013.pdf

But, in that case, what about the electric field (given by Ohm's law) which is driving the current in the first place? That one is certainly not zero. Or does it "converge to zero" since the conductor is defined to be "very long" and ohm's law gives $\lim_{|d| \to \infty} IR = \vec{E} \cdot \vec{d}$

 

[Vagn]

Homework Statement

Problem 1c from here: http://web.phys.ntnu.no/~ingves/Teaching/TFY4240/Exam/Exam_tfy4240_Dec_2013.pdf

Homework Equations

Maxwell's equations

The Attempt at a Solution

According to the solutions, the electric field is ZERO everywhere because it's "magnetostatics" and because the net charge is zero everywhere. http://web.phys.ntnu.no/~ingves/Teaching/TFY4240/Exam/Solution_tfy4240_Dec_2013.pdf

But, in that case, what about the electric field (given by Ohm's law) which is driving the current in the first place? That one is certainly not zero. Or does it "converge to zero" since the conductor is defined to be "very long" and ohm's law gives $\lim_{|d| \to \infty} IR = \vec{E} \cdot \vec{d}$

Isn't the question asking for the radial electric field (i.e. the E-field in the x-y plane), the field driving the current would be along the wire not radially.

 

[Nikitin]

No it's asking for the Electric field $\vec{E}(\vec{r})=\vec{E}(\vec{r_{||}})$, i.e. E as a function of radial r.

 

[Vagn]

No it's asking for the Electric field $\vec{E}(\vec{r})=\vec{E}(\vec{r_{||}})$, i.e. E as a function of radial r.

Part b asks for the magnetic field as a function of distance from the z-axis, where the current flows parallel to z. It then asks you to repeat the process for the electric field, which would imply it is the electric field perpendicular to the current.
In the solutions to part b, the magnetic field is given as a function of the radial component only, the question is similarly explained in the footnote at the bottom of the question page, where it states 'Obtain the electric field E(r) as a function of r_||=|r_|||' for the same regions.

 

[Nikitin]

they asked for the magnetic field as a function of radial distance. i don't think they said anything about finding the magnetic field-component perpendicular to the current.

Regardless, wouldn't the electrical field inside a conducting wire like this approach zero anyway in the limit its length going towards infinity? I mean, electrical field times length equals difference in potential, and if voltage remains constant while length diverges, then the electrical field must approach zero.

 

[vela]

As the wire gets longer, the resistance R would increase, so to keep the current fixed at I, you'd have to increase the potential difference V. In the microscopic view, you'd have $\vec{J} = \sigma\vec{E}$. Neither the current density nor the conductivity depend on the length of the wire, so $\vec{E}$ doesn't depend on the length either.

You're right that in a real wire, there'd be an electric field directed along the wire to keep the charges moving. You're supposed to use the approximation that this is an ideal conductor, so there's no resistance. Once the current is established, no electric field is needed to keep it moving.