Question

find ( i_3(t) )
answer: ( i_3(t) = 1.333 sin t , a )

Explanation:

Step1: Calculate the equivalent resistance of the parallel part

The formula for equivalent resistance \(R_{eq}\) of two parallel resistors \(R_1\) and \(R_2\) is \(\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}\). Here \(R_1 = 6\Omega\) and \(R_2=3\Omega\), so \(\frac{1}{R_{eq}}=\frac{1}{6}+\frac{1}{3}=\frac{1 + 2}{6}=\frac{3}{6}=\frac{1}{2}\), then \(R_{eq}=2\Omega\).

Step2: Calculate the total resistance in the circuit

The total resistance \(R_{total}\) is the sum of the series - connected resistances. The \(4\Omega\) resistor is in series with the equivalent resistance of the parallel part. So \(R_{total}=4 + 2=6\Omega\).

Step3: Calculate the total current in the circuit

Using Ohm's law \(i(t)=\frac{v(t)}{R}\), where \(v(t)=12\sin t\) and \(R = 6\Omega\). So \(i(t)=\frac{12\sin t}{6}=2\sin t\) A.

Step4: Calculate the voltage across the parallel part

The voltage across the parallel part \(v_x\) is equal to the voltage across the equivalent resistance of the parallel part. Using \(v = iR\), with \(i = 2\sin t\) and \(R = 2\Omega\), so \(v_x=2\sin t\times2 = 4\sin t\) V.

Step5: Calculate \(i_3(t)\)

Using Ohm's law \(i_3(t)=\frac{v_x}{R_3}\), where \(v_x = 4\sin t\) and \(R_3=3\Omega\). So \(i_3(t)=\frac{4\sin t}{3}\approx1.333\sin t\) A.

Answer:

\(i_3(t)=\frac{4}{3}\sin t\approx1.333\sin t\) A