Total charge lifted by the charge escalator

[ozma]

total charge lifted by the "charge escalator"

Homework Statement

A 1.5 V flashlight battery is connected to a wire with a resistance of 5.00 ohms. The figure shows the battery's potential difference as a function of time. The slope is linear going from (0, 1.5 V) to (2 hrs, 0).

Homework Equations

I = \DeltaV / R
I = dQ / dt

The Attempt at a Solution

I = 1.5 V / 5 ohms = 0.3 A
0.3 A = dQ / (2 hrs x 60 x 60) --> dQ = 2160 C

This is not the correct answer, and I don't understand why. Thanks for your help.

 

[xcvxcvvc]

Homework Statement

A 1.5 V flashlight battery is connected to a wire with a resistance of 5.00 ohms. The figure shows the battery's potential difference as a function of time. The slope is linear going from (0, 1.5 V) to (2 hrs, 0).

Homework Equations

I = \DeltaV / R
I = dQ / dt

The Attempt at a Solution

I = 1.5 V / 5 ohms = 0.3 A
0.3 A = dQ / (2 hrs x 60 x 60) --> dQ = 2160 C

This is not the correct answer, and I don't understand why. Thanks for your help.

When you solve the problem like that, you are assuming the current was .3A for the entire 2 hours, but as the graph shows, the voltage and therefore current steadily increase with time.

Set the problem up more like this:
\frac{V(t)}{R}=\frac{dQ}{dt}
You can solve it with one integration. If you don't know integration, you can solve it the way you did and use the average voltage.

 

[ozma]

When you solve the problem like that, you are assuming the current was .3A for the entire 2 hours, but as the graph shows, the voltage and therefore current steadily increase with time.

Set the problem up more like this:
\frac{V(t)}{R}=\frac{dQ}{dt}
You can solve it with one integration. If you don't know integration, you can solve it the way you did and use the average voltage.

So I tried this:

average of the voltage (1.5 V) over 2 hours = 1.5 / (2 x 60 x 60) = 2.1 x 10^-4

average I = average V / R = 2.1 x 10^-4 / 5 = 4.2 x 10^-5

average I = dQ / dt --> 4.2 x 10^-5 = dQ / (60 x 2 x 2) --> dQ = 0.01

but this answer is still incorrect. Any ideas? Thanks.

 

[xcvxcvvc]

So I tried this:

average of the voltage (1.5 V) over 2 hours = 1.5 / (2 x 60 x 60) = 2.1 x 10^-4

average I = average V / R = 2.1 x 10^-4 / 5 = 4.2 x 10^-5

average I = dQ / dt --> 4.2 x 10^-5 = dQ / (60 x 2 x 2) --> dQ = 0.01

but this answer is still incorrect. Any ideas? Thanks.

The average value of that function is 1.5/2 = .75 volts.

Edit: You should probably solve it the calculus way since you've been told about the derivative definition of current:
\frac{V(t)}{R}=\frac{dQ}{dt}
First we define V(t) as a function of time. It's of the form mx+b since it's linear, and by looking at the graph, you can calculate m and see that b = 0.

multiply both sides by the differential time:
\frac{mt*dt}{R}=dQ
Integrate both sides:
\int_{t_i}^{t_F}\frac{mt}{R}\, dt=\int_{0}^{Q_{total}} dQ
Q_{total}=\int_{t_i}^{t_F}\frac{mt}{R}\, dt
Q_{total}=\frac{mt^2}{2R}|_{t_i}^{t_F}

 

[ozma]

The average value of that function is 1.5/2 = .75 volts.

Edit: You should probably solve it the calculus way since you've been told about the derivative definition of current:
\frac{V(t)}{R}=\frac{dQ}{dt}
First we define V(t) as a function of time. It's of the form mx+b since it's linear, and by looking at the graph, you can calculate m and see that b = 0.

multiply both sides by the differential time:
\frac{mt*dt}{R}=dQ
Integrate both sides:
\int_{t_i}^{t_F}\frac{mt}{R}\, dx=\int_{0}^{Q_{total}} dQ
Q_{total}=\int_{t_i}^{t_F}\frac{mt}{R}\, dx
Q_{total}=\frac{mt^2}{2R}|_{t_i}^{t_F}

I don't understand why the average V would be 1.5 / 2. Shouldn't I divide by the time in seconds? 2 hours = 2 x 60 x 60 seconds, that's why I used this number as the denominator.

Q_{total}=\frac{mt^2}{2R}|_{t_i}^{t_F}

What numbers do I put in this? Is m the slope of the graph? Would that be 1.5 / 2 or 1.5 / (2 x 60 x 60)? And what am I using for t, 2 or the time in seconds?

If I use t = 2, then I have Q = ((1.5/2) x 2^2) / (2 x 5) = 0.3 C

 

[xcvxcvvc]

I don't understand why the average V would be 1.5 / 2. Shouldn't I divide by the time in seconds? 2 hours = 2 x 60 x 60 seconds, that's why I used this number as the denominator.

Q_{total}=\frac{mt^2}{2R}|_{t_i}^{t_F}

What numbers do I put in this? Is m the slope of the graph? Would that be 1.5 / 2 or 1.5 / (2 x 60 x 60)? And what am I using for t, 2 or the time in seconds?

If I use t = 2, then I have Q = ((1.5/2) x 2^2) / (2 x 5) = 0.3 C

You wouldn't divide by the time to find average voltage. An easy way to check this is to see the units would be volts per second for average voltage... We want volts. The function's average is the middle value of the linear function, because the higher parts past the halfpoint are exactly balanced by the lower parts before it.

As for your second question, if you are integrating, you can use time in hours if the slope is in voltage per hour or you can use seconds if the slope is in voltage per second. It's up to how you find your slope. Slope WOULD be V/s unlike average voltage. m = 1.5/(2*3600) for volts per second.

However, when you're using average voltage, you need to use seconds and cannot use hours or anything due to the definition of current (which is found by V/R). 1.5/2 * 2 * 3600/5 = total charge.

 

[ozma]

i did average V / R = (1.5 / 2) / 5 = 0.15
I = dQ / dt --> 0.15 = dQ / (2 x 60 x 60) --> dQ = 1080 C
and it's correct! finally! thanks for your help.