Two variants of the Right hand rule

[PhysicsTest]

TL;DR Summary

I found the two variants of right hand rule to find the direction of magnetic field. Need clarity on when to apply each rule.

I found two variants of Right hand rule one of the straight wire and to find the direction of the magnetic field i need to use the below method

Method1:

And to find the direction of magnetic field produced by current in a loop

Method2:

In method1 the thumb was pointing in the direction of current, but in Method2 the fingers are curl in the direction of current and the thumb points in the direction of Magnetic field. Why this difference and when to apply the Method1 and Method2?

 

[Kyouran]

It just turns out that for the current loop they're equivalent. Try applying the first method to the loop and you'll realize it's the same. Obviously you can't really apply the second version if there is no loop to curl your fingers. So I'd go with the first version if you have no loop and with the second if you have a loop. But when in doubt, you can always go back to basics...hard math.

 

[Joshy]

They both look the same to me; just one is using the fingers to follow the current and the other uses the thumb.

You can either or both, or use the one that makes you most comfortable and that you can follow consistently.

"back to basics" and "hard math" is fantastic when you have time. I use to forget and get twisted up and I'd go back to basics, but the exams often had a time constraint deriving things during the exam would eat up valuable time and easily pave the way to a failing grade instead of the perfect score I felt like I could have earned. Even when the exam asks you to solve the problem using math the students who have the insight/intuition are likely going to do better. Knowing the material isn't good enough for exams you're being tested on your intuition and if the class is curved, then you're competing for your grade and if other students have intuition or tricks to follow you're going to be running a race on one foot.

 

[Kyouran]

Well, "hard math" is a bit of an exaggeration. Draw a coordinate system, and you let for example the current flow in the positive $\vec{e_z}$ direction and the radial distance vector from the wire is in the $\vec{e_r}$ direction then just use $\vec{e_z} \times \vec{e_r} = \vec{e_{\theta}}$ in cylindrical coordinates. You don't need to use the full Biot-Savart law for it.