The current flowing through a 1000 ohm resistor varies according to the equation i=0.5 exp(-250t).If the power consumed by the resistor is equal to the rate at which the energy is used, how much energy is lost in the resistor from t=0 to t=0.001 sec? Hint:w=∫Ri^2dt (use i^2 not i)

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The current flowing through a 1000 ohm resistor varies according to the equation i=0.5 exp(-250t).If the power consumed by the resistor is equal to the rate at which the energy is used, how much energy is lost in the resistor from t=0 to t=0.001 sec? Hint:w=∫Ri^2dt (use i^2 not i)

Mathematics
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solve the integral in the interval from t=0 to t=0.001
\[I=\int\limits_{}^{}1000(0.5^2e ^{-500t})dt=\int\limits_{}^{}250e ^{-500t}dt\] \[=-0.5e ^{-500t}\] substitute for the interval from t=0 to t=0.001 to get \[-0.5(e ^{-500(0.001)}-e ^{0})=-0.5(e ^{1/2}-1)\] therefor the energy lost in the resistor = 0.5(e^0.5 -1) I hope it's right
My possible answers are; 1) 197*10^(-6) J; 2)197*10^(-3) J; 3)0.86 J; 4)1.234 J ; 5) 1.68 J; 6) 2.756 J (All units are in JOULES )

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Other answers:

oh sorry I forgot the minus in the power
it will be -0.5(e^(-.5)-1)=-0.197 joules so the energy lost is 197*10^(-3)J (answer 2)
sorry for the typo mistake
does that make sense to you?
Yes.Thank you !!!
np

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