Powering Up A Connected Sata Drive After Computer Is Already Booted Up

I have an internal sata drive that has 2 partitions, one of which is a bootable partition. I decided that I want to back up some data to this hard drive. If I connected the drive to my computer straight away, I was concerned that my computer would not know which partition to boot to. The computer whose data I want to back up obviously already has a partition that it boots to. So, I connected the drive that I would like to backup to but I did not plug in the external AC power adapter to the wall. Then I powered up the computer. Once the computer booted up, then I plugged in the power for the backup drive. However, the drive did not come up in Windows explorer.
1) should that normally work?
2) Why isn't it working in my case? I put the drive in another computer and I can definitely boot to it.

 

Hook it up, then turn on your PC. There's a bootloader that tells the BIOS/UEFI which partition to boot from. It won't boot from the other hard drive.

 

Is the UEFI/BIOS set to AHCI or RAID operation mode? It supposedly supports hot plugging.

 

Upon searching through the bios menus, I see that the storage option is set to AHCI. So, seems like what I did should have worked, right? I was pretty careful about everything being plugged in sufficiently (no loose cables).

 

Ok. I will try that in a few days. I am juggling a lot of other work right now so I don't have time.