Find v(t) and i(t) in the RC circuit

[zr95]

Homework Statement

Homework Equations

Voltage Division
v₂ = v_s * (r₂/(r₂+r₁))
KCL - sum of all currents at a given node is 0
V= i * r

The Attempt at a Solution

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This is my solution:
https://scontent-atl3-1.xx.fbcdn.net/v/t34.0-12/14962850_1306124986078874_1009698561_n.jpg?oh=f9056525f1013bcfa49c664feeff07e2&oe=581B4244

This is my professor's solution:

I understand finding v(t) but when it comes to i(t) I'm confused.
First conceptual question:
I know the current across an inductor must be continuous at t = 0 but does this also apply to the current through a capacitor at t = 0? I imagine this isn't the case since neither me or the professor's solution point to this.

Next question, how is 0 mA the answer for t<0 ? If the switch is closed during this time then there should still be a current going down that path.
Was my approach for finding i(t) when t<0 incorrect? I drew the circuit for t<0, under DC regime a capacitor can be treated as an open circuit, I found R_th and found the current of that circuit. Wouldn't the current of that circuit be i(t) for t<0?

I know I asked a lot of questions so answering any part of it, if not all, would still be extremely helpful.
Thanks in advance!

 

[cnh1995]

does this also apply to the current through a capacitor at t = 0?

No. Capacitor current can change instantaneously, which prevents the voltage across the capacitor to change instantaneously. Capacitor resists sudden changes in the voltage across it.

how is 0 mA the answer for t<0 ?

I believe you are expected to assume i(t) as the capacitor current. Hence for t<0, it is 0.