Determinate the direction of the current of current-carrying wire by the right-hand rule

Question

I have experienced a conundrum while using the right-hand rule to determine the direction of the current. I read a textbook that said: "Curl the fingers around the integration path..." However, I don't know why the direction of the current of the second conductor, $I_2$, is downward.

According to the picture, the integration path is counterclockwise, and $\mathrm{d}\mathbf{l}$ is also counterclockwise. Therefore, shouldn't the direction of the current on the second conductor be upward instead of downward?

Answer 1

Each of the wires will generate its own magnetic field and you could calculate those magnetic fields individually, and use the right-hand rule to work out in which direction the magnetic fields circulate (not vice-versa), and then add up the magnetic fields (as vector fields) to work out the net magnetic field at any point.

Ampere's law tells you that you don't necessarily have to do that if you just want the line integral of the B-field. What matters then is just the net current passing through a closed loop. The direction of the magnetic field along that loop will depend on the net current interior to the loop and on the geometry of the loop and how close it passes to the individual wires. Thus in your picure, I would guess that if $I_1 + I_3 > I_2$ then the B-field is likely to circulate counter clockwise (looking down from the top).

A caveat here is that this is not a very symmetric current distribution. The influence of $I_2$ will be greater closer to the wire carrying that current and vice-versa. Ampere's law only tells you the total line integral, not the value or the direction of the field at any point, except in highly symmetric situations.

Answer 2

Ampere's law:

According to Ampere's law, the magnetic field line integral around a closed path is equal to the product of the magnetic permeability of that space and the total current through the area bounded by that path.

Right Hand Thumb Rule:

If a current carrying conductor is imagined to be held in your right hand such that the thumb points along the direction of current, then the direction of the wrapped fingers will give the direction of magnetic field lines. Q.

These two are different things. ..

While here you seem to curl fingers around integral path, you should actually point your thumb towards direction of current for individual wires and to get direction of magnetic field follow the curl of your fingers. While ampere's law is used to know the magnetic field line intergral in a closed loop, the current concerned here will be net current.