A LR circuit has a 10V battery, 5.50H inductor, and a 6.7 Ohm resistor. The battery is closed at t=0. (a) What is the time constant of the circuit? (b) How much energy is delivered by the battery during the first 2 seconds? (c) How much of this energy is stored in the magnetic field of the inductor? (d) How much of this energy is dissipated in the resistor?
The Attempt at a Solution
a) This is simple, time constant = R/L
b)my book never defines an equation for the total work done by the battery excpet dW/dt=IE. When I use this equation with the current value at t=2 (and dt=2), and solve for dW with the initial battery voltage, I get an answer of about 27 Joules. However, I don't think this is right since the next two parts don't agree with this answer.
c)The energy in an inductor is stored in the magnetic field created by the inductor. So UL=1/2LI^2. When using the given values and the current solved for at t=2 i Get around 5.10 Joules.
d) I would assume that if the two previous values are correct, then the difference between the two would be the answer. However, looking for a way to confim this, I did an intergral of I^2R over time. So I had (E/R[1-e^(-tR/L)])^2Rdt and integrated from t=0 to t=2. When I solved for this value I got a little less then 14 Joules.
14+5 != 27. So what's wrong here?