# Finding current and charge in a RC circuit.

• U.Renko
In summary, the equation for q is correct, but you need to solve for q(t) with the initial condition q(0)=0.
U.Renko

## Homework Statement

A resistor with resistance R and a capacitor with capacitance C are connected in series to a direct current battery ε.
find the current and charge on the circuit as function of time.

it looks more like a review of differential equations, so I'm not really sure if I should post here or in the calculus forum...feel free to move it if you think it's better.

## Homework Equations

potential at resistor: $V_r = Ri$

potential at capacitor:$V_c = \frac{q}{C}$

## The Attempt at a Solution

Applying one of the Kirchhoff's law:
$\epsilon - Ri - \frac{q}{C} = 0$
$\epsilon = Ri +\frac{q}{C}$
since $i = \frac{dq}{dt}$ we can rewrite the equation as

$\frac{\epsilon}{R} = \frac{dq}{dt} + \frac{q}{RC}$ and solve with an integration factor $e^{\frac{t}{RC}}$
so we have:
$\frac{\epsilon}{R} e^{\frac{t}{RC}} = \frac{d}{dt}(q e^{\frac{t}{RC}})$
and then:

$\int \frac{\epsilon}{R} e^{\frac{t}{RC}} dt = q e^{\frac{t}{RC}}$

here I'm kinda stuck:
because the problem did not give any initial conditions, should I just solve an indefinite integral or integrate from zero to an arbitrary t??

also: $q e^{\frac{t}{RC}}$ comes from integrating $\int_{a}^{b} \frac{d}{dt}(q e^{\frac{t}{RC}})dt$ and applying the fundamental theorem of calculus.
However, don't I need the values a and b to properly use it?

The resistor and capacitor are connected in series to a battery. You can count the time from the instant when you close the circuit. At that instant you can assume zero charge on the capacitor, if it is not stated otherwise.

The differential equation for q is correct,

$$\frac{\epsilon}{R} = \frac{dq}{dt} + \frac{q}{RC}$$

Solve for q(t) with the initial condition q(0)=0. ehild

1 person
hmm let's see:

$\int_{0}^{t} \frac{\epsilon}{R}e^{\frac{t}{RC}}dt = qe^{\frac{t}{RC}$
solving the left hand side by substitution gives:
$C\epsilon\left(e^{\frac{t}{RC} -1 \right) = qe^{\frac{t}{RC}$
and so:

$q(t)= e^{\frac{-t}{RC}C\epsilon\left(e^{\frac{t}{RC} -1 \right)$

$q(t)= C\epsilon\left(1-e^{\frac{-t}{RC} \right)$
and
$i(t) = \frac{\epsilon}{R}e^{\frac{-t}{RC}$

is that correct?

Last edited:
U.Renko said:
hmm let's see:

$$\int_{0}^{t} \frac {\epsilon}{R} e^{\frac{t}{RC}}dt = q e^{\frac{t}{RC}}$$
solving the left hand side by substitution gives:
$$C\epsilon \left (e^{\frac{t}{RC}} -1 \right) = qe^{\frac{t}{RC}}$$
and so:

$$q(t)= e^{\frac{-t}{RC}}C\epsilon \left (e^{\frac{t}{RC}} -1 \right)$$

$$q(t)= C \epsilon \left(1-e^{\frac{-t}{RC}} \right)$$
and
$$i(t) = \frac {\epsilon}{R}e^{\frac {-t}{RC}}$$

is that correct?

Made some corrections. Are these the equations you meant?

1 person
And if so, then yes, you are correct.

yes, that is what I meant.

I honestly don't know where I messed up with the LaTeX though...

U.Renko said:
yes, that is what I meant.

I honestly don't know where I messed up with the LaTeX though...

Just missed a few brackets here and there. That's enough to break it though haha

## 1. How do you calculate the charge on a capacitor in an RC circuit?

The charge on a capacitor in an RC circuit can be calculated using the formula Q = CV, where Q is the charge, C is the capacitance of the capacitor, and V is the voltage across the capacitor.

## 2. What is the relationship between current and charge in an RC circuit?

In an RC circuit, the current and charge are directly proportional. This means that as the charge on the capacitor increases, the current also increases. Similarly, as the charge decreases, the current decreases.

## 3. How do you find the current in an RC circuit?

The current in an RC circuit can be found using the formula I = V/R, where I is the current, V is the voltage across the circuit, and R is the resistance in the circuit.

## 4. What happens to the current over time in an RC circuit?

In an RC circuit, the current starts at its maximum value and decreases over time as the capacitor charges up. Eventually, the current reaches zero as the capacitor becomes fully charged.

## 5. How do you find the time constant in an RC circuit?

The time constant in an RC circuit can be found using the formula τ = RC, where τ is the time constant, R is the resistance in the circuit, and C is the capacitance of the capacitor. This value represents the time it takes for the capacitor to charge up to approximately 63% of its maximum charge.

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