Question

electricity use the formula v = ci, where v is the voltage, c is the current, and i is the impedance. the voltage in a circuit is 24 - 8j volts, and the impedance is 4 - 2j ohms. what is the current?

Explanation:

Step1: Rearrange the formula for current

Given $V = CI$, we can solve for $C$ as $C=\frac{V}{I}$. Here, $V = 24 - 8j$ and $I=4 - 2j$. So $C=\frac{24 - 8j}{4 - 2j}$.

Step2: Rationalize the denominator

Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $4 - 2j$ is $4 + 2j$.
\[

$$\begin{align*} C&=\frac{(24 - 8j)(4 + 2j)}{(4 - 2j)(4 + 2j)}\\ &=\frac{24\times4+24\times2j-8j\times4-8j\times2j}{4^{2}-(2j)^{2}}\\ &=\frac{96 + 48j-32j-16j^{2}}{16 - 4j^{2}} \end{align*}$$

\]
Since $j^{2}=- 1$, we have:
\[

$$\begin{align*} C&=\frac{96 + 48j-32j-16\times(-1)}{16-4\times(-1)}\\ &=\frac{96 + 16+48j-32j}{16 + 4}\\ &=\frac{112 + 16j}{20}\\ &=\frac{112}{20}+\frac{16j}{20}\\ &=\frac{28}{5}+\frac{4}{5}j \end{align*}$$

\]

Answer:

$\frac{28}{5}+\frac{4}{5}j$