Current Decay in an R-L Circuit

[sdalglish13]

I was trying to solve this problem the other day for my physics class and I keep getting the wrong answer. The problem is as follows:

An ideal solenoid is 18.5 cm long, has a circular cross-section 2.20 cm in diameter, and contains 545 equally spaced thin windings. This solenoid is connected in a series circuit with a 15 ohm resistor, a battery of internal resistance ohms and open-circuit terminal voltage of 25 V, and an open switch. (Note u = 4*(pi)*10^-7 T *m/A)

How long after closing the switch will it take for the stored energy in the solenoid to reach 1/2 of its maximum value?


So this is what I did:

L = (u₀N²A)/l

where L = inductance in Henry
N = number of turns
A = area of cross-section
l = length in meters

L = (4*(pi)*10^-7 )(545²)(3.8*10^-4)
.185

L = 7.66 *10^-4 H

Then I found the current I:

I = emf/R

where emf = electromotive force/voltage
R = resistance

I = 25 Volts/(15 ohms + 5 ohms) = 1.25 Amps

Next I used the energy equation to find the maximum energy:

U = 0.5*L*I²

where U = energy
L = inductance
I = current

U = 0.5*(7.66 *10^-4)*(1.25²)
U = 5.99*10^-4 J


Then I don't know where to go from there to find time. I already tried an equation I found in my textbook...

U = U₀e^-2*(R/L)*t

t = -ln(.5)*L/(2R)
t = -ln(.5)*(7.66 *10^-4)/(2*(15+5))
t = 1.32*10^-5 sec

and solved for t that way, but I keep getting 1.32*10^-5 sec, when the answer should be t = 4.71 *10^-5. I know t = 4.71 *10^-5 is the correct answer because it came off of the answer sheet for a review.

Can anyone shed some light on what I am doing right/wrong?

 

[vela]

You need to find the current as a function of time and use it to calculate the energy stored in the inductor as a function of time.

The equation you tried doesn't apply to this situation. For one, it starts at U₀ and decays to 0, but in this circuit, the energy starts at 0 and increases to the max value you found as t goes to infinity.