How should apathy in the classroom be dealt with?

I see you have chosen to list this question under "Law and Politics," so I will try to focus on teaching in those areas, but really, the means of dealing with apathy in any content area are not all that different.  Apathy can be overcome only with student engagement, and that is the responsibility of the teacher to create.

To engage a student, you must meet the student where he or she is.  Every student is in a different place, cognitively, socioeconomically, and emotionally.  Each has had a different set of experiences.  It is up to the teacher to find out where the student is in all those dimensions and something about the nature of the student's experiences. Students can be enticed into overcoming their apathy if they see that there is a sincere attempt to meet them where they are.  A student may have never read the Constitution.  That student is in a different place from the student whose father or mother carries around a copy of the Constitution. How can you engage someone to be interested in a document he or she has never seen if you don't realize this?  Sometimes apathy is a defense.  The student who is a poor reader may be simply covering up this deficiency with what appears to be apathy.  It is up to us to get to know and understand each of our students.  This might seem like an overwhelming task, but it is doable, simply by paying attention.  Without understanding the great differences amongst students, a teacher can be singularly unengaging.  Understanding our students is what allows us to create tailored strategies of engagement. 

Students are like anyone else; what interests all of us is likely to be something close to home.  A student may not know who represents him or her in city government. But that student may very well get engaged if that representative has voted for an ordinance that has some impact on university students.  This kind of thing is what triggers much citizen engagement.  Most students today have some work experience.  They can be engaged in great First Amendment discussions as this amendment relates to their work experiences or those of their parents.  They may not realize they do not have free speech rights on the job or that they may be entitled to some religious accommodation on the job.  And this is the perfect time to be teaching courses in law and politics, with the present election. Even the most apathetic of students will be aware there is an election going on.  If you know your students, you can engage them in conversation and speculation about how each candidate would affect their lives, vis-a-vis education or employment, for example.  In a southern clime, the failure of Congress to provide for funding to deal with the Zika virus might have some impact on your students' lives. 

Engagement is about making students curious about the world around them, so it only makes sense to start with their particular worlds.  And to do this effectively, we must know what those worlds are and what our students bring to the them.