There isn't a "the" class path. CLASSPATH (all uppercase) is the shell environment variable name that Sun designated for the Java runtime to access in order to get the list of class sources.
Each Linux command shell instance has its own variable environment, although it may simply be a copy of its parent, depending on how that shell was spawned.
Depending on which shell you're using and what implementation of Linux or Unix you're working with, there are multiple places you can setup a "master" CLASSPATH. I normally use bash and set it up per-user, so I define my CLASSPATH in ~/.bash_profile.
Since CLASSPATH is an environment variable, you can set it inside a given shell session to a new value at any time or unset it entirely. That will only affect that shell session unless you export the change out to the shell's parent.
Note that when you invoke a script, it normally creates a new shell instance, so whatever CLASSPATH you set there won't reflect back to the caller unless you export it.
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