New main objective: Test J'Zargo's Flame Cloak spell on the undead
Note! This mission is optional, but well worth completing as you might gain an ally and increase your magicka skills. You will be able to take on this quest no sooner than after finishing the second main quest of the College of Winterhold, i.e. Under Saarthal.
Find J'zargo, one of the College students. The Khajiit can be most of the times met at the Hall of Attainment, though he might also be walking through other locations. Make sure that you used up all the standard dialogue options and J'zargo should ask you for help in testing his magic scrolls (screen above). Khajiit will give you 10 J'zargo's Flame Cloak Scrolls and ask you to test them on the undead.
The undead category most of all includes Draugrs and Skeletons. You can head to any location you like or just be patient and complete the Khajiit's request during the Good Intentions main quest. Approach a group of undead, open your inventory and choose J'zargo's Flame Cloak (screen above). Close the inventory window and BE SURE to tell your party member to say behind, as otherwise you might hurt him pretty badly. Move away from him and use the scroll, causing flames to surround the hero.
Now you just need to approach the monsters and wait for the spell's effects (screen above). Unfortunately J'zargo's scrolls will also have side effects, therefore causing random explosions apart from damaging the undead. Make sure you have some healing potions or spells and regenerate your HP when needed.
New main objective: Speak with J'zargo
After completing the required three tests, return to J'zargo and inform him of the scrolls not working all that well. Unfortunately there's no prize in return for completing this quest, though from now on you will be able to ask the Khajiit to join your party.