Acid arrow and sneak attack damage

Question

For spells that deal damage over time such as acid arrow, Contagious Flame, or Holy Javelin, how does sneak attack damage work with them? This question assumes the attack hits, and all requirements for sneak attack are fulfilled for the first round. Later rounds the qualifying nature of sneak attack can be discussed.

The first round of AA and HJ I think everyone can agree on that the sneak attack damage would be applied because attack roles were made. The debate starts on the 2nd round of damage. Does the continuing damage benefit from sneak attack?

The first round of CF is just like AA, attack roll(s) was made, so the sneak attack gets applied. Now for the second round bouncing attacks. Would these benefit from the original sneak attack damage boost since the spell fires off a new ray each round, or would it benefit from sneak attack from me, or better would the original damage that includes the sneak be used and then sneak gets applied AGAIN (yes I am dreaming, but it is possible). For any of these do I still have to qualify for the rules of sneak attack such as be invisible, or worse, does the target I hit have to somehow meet the requirements?

Ive tried asking a variant of this question before but it was closed here. So I am limiting it to something more specific than the original.

Answer 1

Acid Arrow/Holy Javelin

Damage over time effects do not activate sneak attack because they are not attack rolls. For these spells, you would only get the initial sneak attack. Then, just like poisons for a more archetypal rogue sneak attacker, they would do their regular damage over time effects.

Contagious Flame

As written, Contagious Flame would sneak attack with one of its initial rays as well as one of its bounce rays each turn. This bouncing damage would also apply your normal sneak attack damage once, because the default damage of the ray is still 4d6. One target, denied their dexterity and within 30ft of "you" (probably the creature your spell is bouncing from, ask your GM, but potentially your character), would be subject to 4d6 + sneak attack damage.