Electric current, resistance and circuits

[pat666]

Homework Statement

See attachment pls.
for a) I really need to know if this is right(well I know its right but it may not be what they want??).
for b) i need some pointers for (ii) and (iii)
very grateful if you would check a) and i) too... thanks to all future helpers

Homework Equations

The Attempt at a Solution

a)
Ohm = Volt / Ampere = Volt second / Coulomb
Farad = Coulomb / Volt

Ohm Farad = Volt second / Coulomb * Coulomb/Volt = second
b)
i) U=Q^2/2C
U=(8.1*10^-3)^2/(2*4.62*10^-6)
U=7.1J (Please check this if you can)
ii)P_loss=I²R
no current though - not sure where to go
iii)Help!

 

[collinsmark]

a)
Ohm = Volt / Ampere = Volt second / Coulomb
Farad = Coulomb / Volt

Ohm Farad = Volt second / Coulomb * Coulomb/Volt = second

Yup. That's one way to do it (there are other ways too, btw.)

b)
i) U=Q^2/2C
U=(8.1*10^-3)^2/(2*4.62*10^-6)
U=7.1J (Please check this if you can)

'Looks good to me.

ii)P_loss=I²R
no current though - not sure where to go
iii)Help!

Don't get discouraged. So you have the equation
P_loss = I²R.
But don't forget about Ohm's law,
V = IR. Or, rewriting it, I = V/R.

Can you think of a way to combine those to get rid of the I?

 

[pat666]

ah so P_loss=V^2R
then I still don't have V though?
and iii) I am at a complete loss for

thanks for your help so far.

 

[collinsmark]

ah so P_loss=V^2R

Not quite!

If P_loss = I²R, and I = V/R then,

P_loss = V²/R

where V is the voltage across the terminals of the resistor R.

then I still don't have V though?

You know that C = Q/V. Rearrange and solve for V.

and iii) I am at a complete loss for

Part iii) is really just a rehash of what what you've already done. Start by halving the answer you found in i), and then work your way through it again with the new numbers.

 

[pat666]

ok thanks for that, ill post my answers back for you to check if you don't mind.

 

[pat666]

hey collinsmark, for ii

P_loss=Q^2/(C^2 R)
P_loss=(8.10*〖10〗^(-3) )^2/((4.62*〖10〗^(-6) )^2*850)
P_loss=3616.3W
seems big?

for iii)
U/2=3.55J=Q^2/2C
C=5.73*〖10〗^(-3) C
P_loss=Q^2/(C^2 R)
P_loss=((5.73*〖〖10〗^(-3))〗^2)/((4.62*〖10〗^(-6) )^2*850)
P_loss=1808.16W
I don't think that that is right since it is exactly half of the power dissipated at full charge and i didn't think that the charging v P_loss curve would be linear??

 

[collinsmark]

P_loss=Q^2/(C^2 R)
P_loss=(8.10*〖10〗^(-3) )^2/((4.62*〖10〗^(-6) )^2*850)
P_loss=3616.3W
seems big?

Yeah, it may seem pretty big. But I also think it's pretty correct.

The voltage is really high on this puppy.
V = Q/C = (8.10 x 10^-3 [C])/(4.62 x 10^-6 [F]) = 1753.25 Volts.

P = V²/R = (1753.25 [V])²/(850 [Ω]) = 3616.34 Watts

Everything checks out.

What's not given in this problem (perhaps you have to solve for it in your next step) is the specific RC time constant for this circuit (although part (a) alludes to this). Yes, the initial power is huge! But you'll find that it doesn't last very long at all.

for iii)
U/2=3.55J=Q^2/2C
C=5.73*〖10〗^(-3) C
P_loss=Q^2/(C^2 R)
P_loss=((5.73*〖〖10〗^(-3))〗^2)/((4.62*〖10〗^(-6) )^2*850)
P_loss=1808.16W

Also looks good to me.

I don't think that that is right since it is exactly half of the power dissipated at full charge and i didn't think that the charging v P_loss curve would be linear??

Careful, some things are linear and some things are not. P_loss vs. V is definitely not linear, as you have said. (P is proportional to the square of V). (And nothing here is linear with respect to time -- that might come in the next problem so I won't talk more about that now.)

But using
U = (1/2)CV², and
P_loss = V²/R, or rewriting as $V = \sqrt{P_{loss}R}$,
find the relationship between U and P_loss (i.e. check whether they are proportional to one another). You'll be pleased with the result.

 

[pat666]

thanks for all your help.
I have another question that is concerned with RC time constant.
If you have time could you please give me some pointers with this question
I don't have any actual values for this so I don't know what to do.

 

[collinsmark]

thanks for all your help.
I have another question that is concerned with RC time constant.
If you have time could you please give me some pointers with this question
I don't have any actual values for this so I don't know what to do.

I suspected a question like that was coming.

You should start by looking in your textbook for a relevant equation (or if you know basic differential equations, you can derive the relationship yourself -- but I'm guessing your textbook has the relationship already in there for you). Then give it a shot at solving the problem. Per the forum rules, we can't help unless you demonstrate that you've attempted to solve the problem (including showing your work).

(By the way, e is the mathematical constant 2.71828... The natural log of e is 1. The solution to this problem involves "e to the power of something" and also performing natural logarithms.)

 

[louza8]

Hey guys can you explain this formula please? P_loss=Q^2/(C^2 R) I don't know where it comes from/what it means :'(

The way I went about this question was the the same to find energy but then diverged so:

-found energy and halved it, rearranged to find charge Q=5.3*10^-3C
-then used the discharging in an RC circuit formula to find the time taken (t) to reach the above charge.
-then used deltaq/deltat (8*1*10^-3 - 5.727*10^-3)/0.00168s to find the current I (1.41)
-then used power dissipation by resistor I^2*R to find power loss

I came up with a different answer like 1700W. Can you guys tell me what assumptions I have made incorrectly? I know I diverged by trying to find current, but I didn't know any other way to get the dissipation.

 

[pat666]

hey,
q(t)=q_max (1-e^(-t/RC) )
t=C*ln(-q_max/(q-q_max))
is q_max something? I am pretty sure that this should go down further but i can't think of what else to sub in?

 

[pat666]

Not quite!

If P_loss = I²R, and I = V/R then,

P_loss = V²/R

where V is the voltage across the terminals of the resistor R.

You know that C = Q/V. Rearrange and solve for V.

Part iii) is really just a rehash of what what you've already done. Start by halving the answer you found in i), and then work your way through it again with the new numbers.

thats where the formula comes from

 

[louza8]

I took q_max to be the initial charge stored in the capacitor 8.1*10^-3, and q(t) to be the amount of charge stored in the capacitor at the half-energy point. RC is given in the question. I was solving for t so I could get current to use in my P_loss formula (I^2R.

I just don't know if I can do it this way (i'm guessing not?), but want to know why I can't/where am I making mistakes.

Finding electricity stuff heaps harder than mechanics!

 

[pat666]

how did you know q at half energy

 

[pat666]

dont worry i see how you knew (exactly how I knew) - not sure why it doesn't work, I am sure someone will enlighten us.

 

[louza8]

On collecting some rounding errors by using my calculator to solve t I'm getting 1700W as the final answer using an I = 1.41 found from deltaq/deltat. Maybe I should write it all out and scan it in or something so you guys can find what I'm doing wrong?

edit:yeah hopefully

 

[pat666]

I don't know if I can tell where your going wrong (I could be wrong and you right) I would be interested in seeing what your doing though.

 

[louza8]

Okay, attached is my part for part iii) but as mentioned i did it a different way. Hope someone can find where I have gone wrong (I think I have assumed something that isn't correct).

-Does the rate of change of current vary with respect to time when discharging from a capacitor? and is that why my deltaq/deltat isn't working? Because they're supposed to be taken in the limit as deltat approaches zero (the instantaneous rate of change) and I've used an average over the 0.00168s period? I'm looking at a graph of current vs time and it appears it doesn't discharge at a linear rate.
Have I answered my own question why my method hasn't worked?

 

[pat666]

Yeah I am thinking what you said could be right. I is C/s how many electrons per 1 sec and when charging or discharging a cap it is certainly not a linear curve. sorry for the late reply - was playing squash - no one on PF reply at this time prbly all americans

 

[collinsmark]

Hello louza8,

Part of your calculations show:

$$\ln \left( {\frac{5.727 \ \mathrm{x} \ 10^{-3} \ [\mathrm{C}]}{8.1 \ \mathrm{x} \ 10^{-3} [\mathrm{C}]}} \right) (850 \ [\Omega]) (4.62 \ \mathrm{x} \ 10^{-6} [\mathrm{F}]) = -t$$

$$t = 0.00168 \ [\mathrm{s}]$$

Check your math. Something is not right with your final value of t.

Then later you assume that

$$I = \frac{\Delta Q}{\Delta t}$$

But your following calculations assume that charge Q is linear with time t. But I assure you it is not! Charge Q is an exponentially decaying function with time!

But if you must take this approach (which I don't necessarily recommend doing by the way -- pat666's solution is much easier), you can do it by finding the instantaneous current.

You already know that

$$q(t) = q_0 e^{\frac{-t}{RC}}$$

So take the derivative with respect to t,

$$\frac{d}{dt}q(t) = -I(t) = q_0 \frac{d}{dt} \left( e^{\frac{-t}{RC}} \right)$$

then use that to find the instantaneous current at time t_1/2 (but again, you'll have to fix your calculation of this t_1/2 value, because there is a mistake in your previous calculation of it, above).

Finally, plug that into P_loss = I²R, and you'll get the correct answer.

But as I said before, although this works, there is a much easier way to handle this problem that doesn't involve calculating any derivatives or anything.

 

[louza8]

Thanks heaps collinsmark, explains it thoroughly.