Can't find correct expression for discharging RL circuit

[Rijad Hadzic]

Homework Statement

After a switch is left for a for many time constants, it is switched to b. find an expression for the total charge that passes through the resistor in one time constant

Homework Equations

dq/dt = I(t)
I(t) = (ε/R)(e^(-tR/l))

The Attempt at a Solution

so dq = I(t) dt
I know the constant is tau but I can't find that so I am just going to call it v, v = l/R

Q = ∫(ε/R)(e^(-t/v)) dt
Q = (ε/R) ∫(e^(-t/v)) dt

let u = -t/v let du = -1/v dt

-du*V = dt

∫(e^(-t/v)) = -v ∫e^u du

∫e^u = e^u

e^u = e^(-t/v) from t to 0

∫(e^(-t/v)) = ε/r * [ -v * (e^(-t/v) - e) ] = Q(t)

its asking for total charge at one time constant (v)

ε/r * [ -v * (e^(-v/v) - e) ] = Q(v)

ε/r * [ -v * (e^(-1) - e) ] = Q(v)

((εv)/(r)) * [ -(1/e) + e ] = Q(v)

but my book is telling me the answer is

((εv)/(r)) * [ (e-1) / e ] = Q(v)
what am I doing wrong?

 

[Eclair_de_XII]

ε/r * [ -v * (e^(-1) - e) ] = Q(t)

((εv)/(r)) * [ (1/e) - e ] = Q(v)

but my book is telling me the answer is

((εv)/(r)) * [ (e-1) / e ] = Q(v)

Did you distribute the negative sign from the first line of this post to the second line?

 

[Rijad Hadzic]

Did you distribute the negative sign from the first line of this post to the second line?

sorry my apologize! I forgot to write that

fixing it up:

ε/r * [ -v * (e^(-1) - e) ] = Q(v)

((εv)/(r)) * [ -(1/e) + e ] = Q(v)

I still don't understand how they got to

((εv)/(r)) * [ (e-1) / e ] = Q(v)

though.

I will edit my OP with the correction that you pointed out

 

[cnh1995]

After a switch is left for a for many time constants, it is switched to b. find an expression for the total charge that passes through the resistor in one time constant

What are a and b? Post the circuit diagram.

 

[Rijad Hadzic]

 

[robphy]

When you evaluate the limits, what term is associated with t=0?

 

[Rijad Hadzic]

When you evaluate the limits, what term is associated with t=0?

Do you mean when I take the integral?

so from

"e^u = e^(-t/v) from t to 0"

you get e^(-t/v) - e^0 = e^(-t/v) - 1

so wait this isn't what I wrote in my OP :S damnit

 

[Rijad Hadzic]

When you evaluate the limits, what term is associated with t=0?

Thanks so much. I'm not sure why I'm making such simple mistakes but I feel like accelerating off a cliff rn.