A 200 -volt electromotive force is applied to an R C -series...

A 200 -volt electromotive force is applied to an $R C$ -series circuit in which the resistance is 1000 ohms and the capacitance is $5 \times 10^{-6}$ farad. Find the charge $q(t)$ on the capacitor if $i(0)=0.4 .$ Determine the charge and current at $t=0.005 \mathrm{s}$ Determine the charge as $t \rightarrow \infty$

Key Concepts

Steady State Behavior

The steady state of an RC circuit refers to the condition as time tends to infinity, where the charging process completes. In a charging circuit, the capacitor becomes fully charged, reaching a voltage equal to the applied electromotive force, and the current in the circuit decays to zero. This behavior represents the equilibrium state of the circuit.

Initial Conditions

In solving the differential equation for an RC circuit, initial conditions such as the initial charge on the capacitor or the initial current are necessary to fully determine the unique solution. They ensure that the mathematical model accurately reflects the physical state of the circuit at the moment the voltage is applied.

Differential Equation for Charging

The charging process in an RC circuit is described by a first-order linear differential equation derived from Kirchhoff’s Voltage Law (KVL). This equation relates the applied voltage to the voltage drop across the resistor and the capacitor and its solution typically involves an exponential function that describes the transient behavior of the system.

Time Constant

The time constant (?) in an RC circuit is calculated as the product of the resistance (R) and capacitance (C) (? = RC). It is a crucial concept because it characterizes the rate at which the capacitor charges or discharges—after a time period equal to one time constant, the voltage across the capacitor typically reaches about 63.2% of its final value.

RC Circuit

An RC circuit consists of a resistor and a capacitor connected together, where the dynamics of how the capacitor charges or discharges when exposed to an applied voltage are of primary interest. The interplay between the resistor and capacitor defines how quickly the capacitor reaches its final charge and how the current decays over time.

Transcript

0:00 Hello.

00:01 So here we suppose that a 200 volt electromotive force is applied to an rc series circuit in which the resistance is 1 ,000 oms.

00:11 And the capacitance is five times 10 to the negative 6th here.

00:20 So we're going to find the charge q of t on the capacitor if i of 0 is equal to 0 .4.

00:27 And then determine the charge and current at t is equal to 0 .0 .0.

00:32 And determine the charge then with st approaches infinity.

00:36 So the differential equation associated with the rc series circuit is going to be r times dqdt plus 1 over cq is equal to eft.

00:46 So here we have that putting in our rc and e of t, we get that 1 ,000 dqdt plus 1 over 5 times 10 to the negative 6q is equal to 200, which then gives us that dq dt plus 200 q is going to be equal to one -fifth.

01:14 So now we have a linear differential equation in q.

01:19 And the integrating factor is then going to be, well, e raised to the integral of 200d, t.

01:26 So we get e to the 200 t.

01:28 So multiplying then by the integrating factor, we have here q.

01:35 Times e to the 200t is then going to be equal to one -fifth times the integral of e to the 200 -t -d -t, which then gives us that q of t is going to be equal to 0 .001 plus c times e to the negative 200t.

02:06 Okay, and now we note that i is equal to dqdt.

02:11 So we get that i of t is going to be equal to negative 200 times c times e to the negative 200 t.

02:19 But we have that i of 0 is equal to 0 .4, which then implies that c is going to be equal to negative 0 .4 over 200...

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