Jackson, Problem 5.2 (b); solenoid currents and fields

[BREAD]

Homework Statement

Homework Equations

The Attempt at a Solution

These are problem 5.1 (b) and the solution.
I understood the gap between each loop is 1/N, but why does each loop have vertical direction current amount of I/N ?? [/B]

 

[Andrew Mason]

The Attempt at a Solution

These are problem 5.1 (b) and the solution.
I understood the gap between each loop is 1/N, but why does each loop have vertical direction current amount of I/N ?? [/B]

The explanation appears to me to be backward. I would look at it this way: The rate at which charge flows in the direction of the solenoid axis is I, the current in the wire. It does not matter how many coils you look at. The current in the direction of the axis is always I so long as the coils continue to progress in the same direction. If the windings were to reverse direction at some point with the end of the coil wire at the same position as the beginning of the wire, there would be no current in the axial direction.

AM

 

[stevendaryl]

Homework Statement

View attachment 203362
View attachment 203363

Homework Equations

The Attempt at a Solution

These are problem 5.1 (b) and the solution.
I understood the gap between each loop is 1/N, but why does each loop have vertical direction current amount of I/N ?? [/B]

Jackson is talking about the case ("a real solenoid") of a continuous coil of wire, which means that the loops are connected. For the loops to be connected, it means that traveling around one loop takes you up a distance of 1/N to get to the next loop. So the velocity of a charge flowing through the loop must have components $v_{tangential}$ and $v_{vertical}$ so that $\frac{v_{tangential}}{v_{vertical}} = \frac{2 \pi R}{\frac{1}{N}}$. Traveling $2\pi R$ tangentially (around the loop) takes place in the same time as traveling $\frac{1}{N}$ vertically (where $R$) is the radius of the loop).

 

[stevendaryl]

The explanation appears to me to be backward. I would look at it this way: The rate at which charge flows in the direction of the solenoid axis is I, the current in the wire.

Well, it's not completely obvious. I is the current through each little bit of wire, but the direction of the current is not vertical. (I'm imagining the coil oriented with the axis vertical). So it's not immediately obvious that there is a current of I in the vertical direction. Or at least not to me.

Here's a way to see it that works for me:

By definition, the total current in the vertical direction at a particular height is the charge per unit time flowing past the horizontal plane at that height. As long as the plane only intersects a single coil of the wire, the charge per unit time flowing past the horizontal plane will be the same as the current I in each coil. It doesn't matter how the wire is coiled, so long as a horizontal plane never intersects more than a single strand of wire. If the wire doubles back in some strange way, then a horizontal plane will intersect more than one strand, and the charge per second through that plane will be something other than I.

So it doesn't matter whether the coil is regular loops, or some random pattern, as long as traveling along the wire always takes you some nonzero distance vertically.

 

[Andrew Mason]

Well, it's not completely obvious. I is the current through each little bit of wire, but the direction of the current is not vertical. (I'm imagining the coil oriented with the axis vertical). So it's not immediately obvious that there is a current of I in the vertical direction. Or at least not to me.

My reasoning is similar to yours except that I just took two arbitrary points on the wire (ie separated by a distance along the axis) and asked: what is the charge per unit time that flows between those points?

AM

 

[Fred Wright]

I suggest that you model one current loop as a helix (the case in a real solenoid) and compute the B field at half the length of the loop using the law of Bio-Savart. You will find that there is a $B_z$ component and a $B_\theta$ component (which is proportional to $B_z$ with proportionality constant equal to the pitch of the helix). Averaging over N loops yields a $B_\theta$ which is the field produced by a linear wire in the z direction.