Energy Stored in Capacitor at t=870μs: Calculations & Answers

[carpaltunnel]

Homework Statement

The initial voltage on the 0.5 microF capacitor shown in the figure is -20 V. the capacitor current has the waveform shown in the figure. https://i.imgur.com/P8jwI7s.jpg?1

How much energy is stored in the capacitor at t = 870 micro seconds.

Homework Equations

v(t) = 1/c * integral from t to t0 i dtao + v(t0) EDIT: "integral from t to t0" is a definite integral with t on the top and t0 on the bottom
w(t) = 1/2cv^2

The Attempt at a Solution

got v(t) = 50*e^-2000t + 30

after plugging into w(t) tested the following answers:

https://imgur.com/3P8ipf4

 

[haruspex]

Your integral calculated the change in voltage (except, you have a sign error) How does that relate to the change in energy?

 

[carpaltunnel]

"from t to t0" is admittedly ambiguous but i mean that t is on the top of the definite integral and t0 is on the bottom. voltage relates to energy by the equation w(t) = 1/2cv^2. i plugged in my equation for v(t) and the time as 870 microseconds. please let me know what else is unclear/wrong. thank you for your reply.

 

[haruspex]

got v(t) = 50*e^-2000t + 30

What does that give for t=0?

voltage relates to energy by the equation w(t) = 1/2cv^2.

But that's different from change in voltage and change in energy.

 

[carpaltunnel]

"what does that give for t=0?"

80 volts. what are you implying?

"but that's different from change in voltage and change in energy"

what are you implying? the expressions for v(t) and w(t) i have provided are the capacitor voltage and energy stored in a capacitor, respectively, at a particular instance for a time. i don't think the answer is w(870 microseconds) - w(0 seconds). please let me know if i am wrong. thank you for your reply.

 

[haruspex]

80 volts.

But:

The initial voltage on the 0.5 microF capacitor shown in the figure is -20 V.

i don't think the answer is w(870 microseconds) - w(0 seconds)

Yes, you're right, I misread the question.