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This is a very basic question about equations but reading online courses about equations didn't clear up what I'm missing.

TLDR; How to get from $T = RC$ to $R = \frac{T}{C}$


Working on a Raspberry PI project where I need to determine the value of a resistor, I decided to use the the step response approach.

I found the the time T in seconds to charge a capacitor with a capacitance of C in Farads and a resistor with a resistance of R in Ohms could be determined using the following equation: $T = RC$

I need to able to calculate the resistance needed for a given T so I tried to swap the equation as follow

$T = RC$

$\frac{T}{R} = \frac{RC}{R}$

$\frac{T}{R} = C$

$\frac{T}{R} * T = CT$

$R = CT$

But considering the fact that I can't calculate R with this equation using the T given by $T = RC$ I obviously made a mistake somewhere.

After some messing around in excel I managed to get the correct equation by using $R = \frac{T}{C}$ but I can't understand how to mathematically swap between $T = RC$ and $R = \frac{T}{C}$

While I understand why dividing the time by the capacitance makes sense to find the resistance, I am very frustrated to not to be able to swap the equation correctly.

Question: Where am I going wrong in my reasoning ?

I haven't used proper calculus in years, so it's probably pretty obvious ... I feel like I should have a $\frac{1}{T}$ somewhere in there but I can't tell why not where ...

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  • $\begingroup$ Welcome to Math SE. I'm not quite sure how you got from the line "$\frac{T}{R} * T = CT$" to "$R = CT$" since $\frac{T}{R} * T = \frac{T^2}{R}$, not $R$ as implied in the next line. $\endgroup$ – John Omielan Aug 18 at 0:51
  • $\begingroup$ Ah ! So that's where I'm going wrong. Can you give me a pointer on how to reduce $\frac{T}{R}$ to just R ? Now that you point it out it seems obvious that multiplying T by T would give T squared ... I believe dividing by T would give $R = \frac{C}{T}$ which still doesn't work... $\endgroup$ – Mathieu VIALES Aug 18 at 1:02
  • $\begingroup$ I'm not quite sure what you're asking & trying to do. What do you mean by "reduce"? Note that $\frac{T}{R}$ (which is $C$ based on your equation) and $R$ are quite different generally in terms of values, and even in terms of dimensions. $\endgroup$ – John Omielan Aug 18 at 1:07
  • $\begingroup$ I'll rephrase my question: How do I get an equation to calculate R using T and C from the equation $T = RC$ ? (Sorry for my improper use of "reduce", being french I'm not really used to the proper English terminology when it comes to calculus)(also, thanks for you time) $\endgroup$ – Mathieu VIALES Aug 18 at 1:10
  • $\begingroup$ From $T = RC$, or the same $RC = T$, dividing both sides by $C$, you have $R = \frac{T}{C}$. This is what you asked about originally. Next, just substitute the values of $T$ and $C$ in the right hand side, i.e., divide the value of $T$ by that of $C$, to get $R$. Does this make sense or am I misunderstanding something? $\endgroup$ – John Omielan Aug 18 at 1:13
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You have the equation

$$T = RC \iff RC = T \tag{1}\label{eq1}$$

To get $R$ by itself, divide both sides by $C$. Since $\frac{C}{C} = 1$, it "disappears" from the left side. Thus, you get

$$\frac{RC}{C} = \frac{T}{C} \implies R\left(\frac{C}{C}\right) = \frac{T}{C} \implies R = \frac{T}{C} \tag{2}\label{eq2}$$

With $T$ and $C$, you can substitute them into the right hand side, i.e., divide $T$ by $C$, to get $R$.

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