RCL circuit, how do I transform the following expressions?

Hey, I don't really understand how you came up with the expression highlighted in red. They're probably basic Maghe rules, but I'm confused. And here I'm having a really hard time transforming it into real and imaginary parts. It's been a constant stumbling block for some reason… 🙁

Double slit?

Hi, I'm not sure how to solve this problem. You can see my approach below, but I'm not sure if it's correct in any way. For me, L is the distance to the screen, and dk is the distance between two adjacent maxima.

Calculate phase shift, how do you get the real and imaginary parts?

Hey, during the semester we had to calculate the phase shift between the output and input voltages. I know that you can calculate the phase shift by dividing the arctan of the imaginary part by the real part of the resistance. But I'm wondering how I know what my Im and Re parts are. The…

Magnetic coil with core?

Hey everyone, I'm sorry that I have so many tasks at the moment, but it's exam time again 🙁 I've found a way to calculate the following problem, but I'm extremely unsure, especially about b. And regarding d: since the susceptibility is less than 1, bismuth should be diamagnetic, right?