Rate of change: q = CVe^(-1/CR); find dq/dt, d^2q/dt^2

[Anthonyk2013]

This is the question.

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3. The charge q on the plates of a capacitor is given by:

. . . . .\(\displaystyle q\, =\, CVe^{-\frac{t}{CR}}\)

where t is the time, C is the capacitance, and R is the resistance. Determine:

(a) the rate of change of charge, which is given by\(\displaystyle \, \dfrac{dq}{dt},\,\) and

(b) the rate of change of current, which is given by \(\displaystyle \, \dfrac{d^2q}{dt^2}.\)

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This is my attempt at solution.

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\(\displaystyle q\, =\, CVe^{-\frac{1}{CR}}\)

\(\displaystyle u\, =\, CV\). . . . .\(\displaystyle \dfrac{du}{dv}\, =\, 0\)

\(\displaystyle v\, =\, e^{-\frac{1}{CR}}\). . . . .\(\displaystyle \dfrac{dv}{dR}\, =\, -\dfrac{1}{CR}\, e^{-\frac{1}{CR}}\)

\(\displaystyle \dfrac{du}{dq}\, =\, \bigg(\, CV\, \bigg)\, \bigg(\, -\dfrac{1}{CR}\, e^{-\frac{1}{CR}}\, \bigg)\, +\, \bigg(\, e^{-\frac{1}{CR}}\, \bigg)\, \bigg(\, 0\, \bigg)\)

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Wondering if I'm on the right track.

 

[stapel]

3. The charge q on the plates of a capacitor is given by:

. . . . .\(\displaystyle q\, =\, CVe^{-\frac{t}{CR}}\)

where t is the time, C is the capacitance, and R is the resistance. Determine:

(a) the rate of change of charge, which is given by\(\displaystyle \, \dfrac{dq}{dt},\,\) and

(b) the rate of change of current, which is given by \(\displaystyle \, \dfrac{d^2q}{dt^2}.\)

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This is my attempt at solution.

\(\displaystyle q\, =\, CVe^{-\frac{1}{CR}}\)

\(\displaystyle u\, =\, CV\). . . . .\(\displaystyle \dfrac{du}{dv}\, =\, 0\)

\(\displaystyle v\, =\, e^{-\frac{1}{CR}}\). . . . .\(\displaystyle \dfrac{dv}{dR}\, =\, -\dfrac{1}{CR}\, e^{-\frac{1}{CR}}\)

\(\displaystyle \dfrac{du}{dq}\, =\, \bigg(\, CV\, \bigg)\, \bigg(\, -\dfrac{1}{CR}\, e^{-\frac{1}{CR}}\, \bigg)\, +\, \bigg(\, e^{-\frac{1}{CR}}\, \bigg)\, \bigg(\, 0\, \bigg)\)

You've been directed to find the first and second derivatives of "q" with respect to "t". You appear to have differentiated one part of the original product with respect to "R" and another with respect to "v" (or "V"?). How do these relate to the original question?

Also, how does "t" relate to the original equation? (There is no "t" in the equation, is why I ask.) Should perhaps there be a "t" in the numerator of the power on the exponential? :wink:

 

[Anthonyk2013]

You've been directed to find the first and second derivatives of "q" with respect to "t". You appear to have differentiated one part of the original product with respect to "R" and another with respect to "v" (or "V"?). How do these relate to the original question?

Also, how does "t" relate to the original equation? (There is no "t" in the equation, is why I ask.) Should perhaps there be a "t" in the numerator of the power on the exponential? :wink:

Ya I didn't spot that about "t", I copied it straight from John birds book.

 

[stapel]

Ya I didn't spot that about "t", I copied it straight from John birds book.

Okay; I don't know what book you're talking about. I'll guess that the original equation, displayed on many websites, is along the lines of the following:

. . . . .\(\displaystyle q(t)\, =\, CV\, \left(\,1\, -\, e^{-\dfrac{t}{CR}}\,\right)\)

I willl guess that the instructions told you to treat C, V, and R as constants. And the instructions clearly state to find the first and second derivatives of "q" with respect to the variable "t".

If I have mis-guessed the text of, and instructions for, the original exercise, please reply with explicit corrections. If I have guessed correctly, then please reply showing your derivatives with respect to the specified variable. Thank you!

 

[Anthonyk2013]

Okay; I don't know what book you're talking about. I'll guess that the original equation, displayed on many websites, is along the lines of the following:

. . . . .\(\displaystyle q(t)\, =\, CV\, \left(\,1\, -\, e^{-\dfrac{t}{CR}}\,\right)\)

I willl guess that the instructions told you to treat C, V, and R as constants. And the instructions clearly state to find the first and second derivatives of "q" with respect to the variable "t".

If I have mis-guessed the text of, and instructions for, the original exercise, please reply with explicit corrections. If I have guessed correctly, then please reply showing your derivatives with respect to the specified variable. Thank you!

Think I know what happened, I posted an image from john bird higher engineering mathematics. The image has been deleted and replaced with text, incorrect text.

As you will see from my reposted image you are one hundred percent right about t, I haven't time at the moment to try the question as kids driving me mad so ill post an attempt later. Thanks for you help~