Current Decay in an R-L Circuit

In summary: To find the time it takes for the stored energy to reach half its maximum value, you can use the equation:t = (L/(2R))*ln(1+(R/(u0*N*A))*I0/Imax)where t is the time in seconds, L is the inductance in Henry, R is the resistance in ohms, u0 is the permeability of free space (4*(pi)*10-7 T*m/A), N is the number of turns, A is the area of cross-section in square meters, I0 is the initial current in amps, and Imax is the maximum current in amps.In summary, to find the time it takes for the stored energy in the solenoid
  • #1
sdalglish13
1
0
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?
 
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  • #2
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.
 

1. What is current decay in an R-L circuit?

Current decay in an R-L circuit refers to the gradual decrease in the flow of electric current in the circuit due to the presence of both resistance (R) and inductance (L).

2. What causes current decay in an R-L circuit?

Current decay in an R-L circuit is caused by the inductive reactance (XL) of the inductor and the resistive losses (R) in the circuit. As the current flows through the circuit, it encounters resistance and inductance, which cause the current to decrease over time.

3. How can current decay in an R-L circuit be calculated?

The rate of current decay in an R-L circuit can be calculated using the formula I = I0e^(-t/τ), where I0 is the initial current, t is the time, and τ is the time constant of the circuit. This formula is derived from Ohm's law and the equation for the current in an RL circuit, I = (E/R)(1-e^(-Rt/L)).

4. How does the presence of inductance affect current decay in an R-L circuit?

The presence of inductance in an R-L circuit causes the current to decrease more slowly compared to a circuit with only resistance. This is because the inductor stores energy in the form of a magnetic field, which opposes the change in current and slows down the rate of decay.

5. What are some applications of understanding current decay in an R-L circuit?

Understanding current decay in an R-L circuit is important in various fields such as electrical engineering, physics, and telecommunications. It is used in designing and analyzing circuits, as well as in the development of devices such as transformers and motors. It is also relevant in the study of electromagnetic waves and their propagation.

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