# Swapping charge time for a capacitor doesn't behave as expected

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 ...

• 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. – John Omielan Aug 18 at 0:51
• 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... – Mathieu VIALES Aug 18 at 1:02
• 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. – John Omielan Aug 18 at 1:07
• 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) – Mathieu VIALES Aug 18 at 1:10
• 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? – John Omielan Aug 18 at 1:13

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