Summary:
There could be scenarios where you would want to assign a User Defined Field (UDF) only to the users of one particular department.

Resolution:
Create a new permission preferably with the name of the UDF suffixed to it (to differentiate the permission).
Create the permission based on the existing permission which has been assigned to users.(This makes sure that all the existing permission's features are available in the new permission)
Enable the new user defined field in the new permission. 
Assign the new permission to users who should be using the new UDF.
Remove the old permission from the users' profile.

Example:
Create a Entire Timesheet type UDF called Finance.
Create a new permission called Timesheet – Finance UDF and enable the Finance UDF in this permission.
You may then assign the Timesheet – Finance UDF permission only to the users who should be using this UDF.