Q. A Yeshiva Bochur accidentally dropped his Tefilah Shel Rosh (out of the plastic box etc.). What is the proper Hanhaga nowadays, as he presumably would not be able to fast whilst adhering to his regular Sedarim in Yeshiva?

A. Mishna Berura (40: 3) writes that the custom is for one who’s tefilin fell on the ground to fast, when not in their bag or protective box.
Chaim Shoal (1: 12), maintains that the aged or ill, can redeem the fast with donating tzedaka. He also mentions that the ones dedicated to learning Torah, when the fast will affect their learning, should rather keep a taanis dibur (restrain from speaking non- Torah conversations) for a day. Others, recommend learning additional time, while some assert that fasting only half a day would suffice.
Divrei Chaim (end of v. 2) explains that when donating tzedaka, it is at least the value of the food he would have saved had he fasted.
Horav Shlomo Miller’s Shlit’a opinion is that a Yeshiva Bochur who has difficulty fasting while maintaining his learning schedule, should only fast half a day and eat before aalos hashachar (daybreak), and also give tzedaka.
If that would still affect his learning, he should redeem the fast with tzedaka. The Rov added that he should also change his ways when handling tefilin, adding especial care, to prevent them from falling in the future.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a