Yes, provided the drive is not being used for any other purpose. If you are then you need to partition it so Time Machine can use the new partition. If you can make the partition at least twice as large as the size of Used space on the drive you are backing up. You should also learn about Time Machine snapshots made automatically and stored on your computer's drive when Time Machine is enabled but the backup drive is not available. See About Time Machine local snapshots.
Get Correct Storage Information
Do not use the information from the Storage section of the About This Mac dialog. Ignore the Storage information as it is typically wrong. To find out the correct information for any disk: Select a Desktop disk icon. Press Command-I to open the Get Info window and look at the topmost panel displayed. You will find the disk information displayed for Capacity, Available, and Used. If you have more than one disk/partition then repeat for each one on your Desktop.
The categories found in the Storage section of About This Mac is simply an arbitrary way of displaying files on your drive. There are no such categories actually on the drive.
Time Machine Snapshots
Open the Terminal in the Utilities folder and enter or paste the appropriate command line. Press RETURN and enter your admin password when prompted. It will not be echoed. Press RETURN again.
To turn them ON: sudo tmutil enable
To turn them OFF: sudo tmutil disable
Note that turning them OFF will also delete all existing snapshots. For more about snapshots see: OS X Mountain Lion- About local snapshots.
There is no better way although there are other ways. However, Time Machine will continue to backup at home once the backup drive is reconnected to the computer.
