Question

the relationship between voltage, \\(e\\), current, \\(i\\), and resistance, \\(z\\), is given by the equation \\(e = iz\\). if a circuit has a current \\(i = 3 + 2i\\) and a resistance \\(z = 2 - i\\), what is the voltage of the circuit?

\\(a\\) \\(4 - i\\)
\\(b\\) \\(4 + i\\)
\\(c\\) \\(8 - i\\)
\\(d\\) \\(8 + i\\)

Explanation:

🆕 New Concept Discovered: Operations with Complex Numbers
Multiplying complex numbers using the imaginary unit \( i^2 = -1 \).

Step 1: Set up the equation

We are given the formula for voltage:

$$ E = IZ $$

Substitute the given values for current \( I = 3 + 2i \) and resistance \( Z = 2 - i \):

$$ E = (3 + 2i)(2 - i) $$

Step 2: Expand the expression

Multiply the binomials using the FOIL method (First, Outer, Inner, Last):

$$ E = (3)(2) + (3)(-i) + (2i)(2) + (2i)(-i) $$

$$ E = 6 - 3i + 4i - 2i^2 $$

Step 3: Simplify using \( i^2 = -1 \)

Combine the like terms containing \( i \):

$$ -3i + 4i = 1i = i $$

Substitute \( i^2 = -1 \) into the expression:

$$ E = 6 + i - 2(-1) $$

$$ E = 6 + i + 2 $$

$$ E = 8 + i $$

Answer:

\( 8 + i \)