To determine the voltage of the circuit, we need to use the given formula:
[tex]\[ \text{voltage} = \text{current} \times \text{resistance} \][/tex]
Given:
- The current is [tex]\( 3 + 4j \)[/tex] amps
- The resistance is [tex]\( 4 - 6j \)[/tex] ohms
Let's denote the current as [tex]\( I = 3 + 4j \)[/tex] and the resistance as [tex]\( R = 4 - 6j \)[/tex].
First, we perform the multiplication of the complex numbers using the distributive property of multiplication over addition for complex numbers:
[tex]\[ (3 + 4j) \times (4 - 6j) \][/tex]
We need to multiply each term in the first complex number by each term in the second complex number:
[tex]\[ (3 \times 4) + (3 \times -6j) + (4j \times 4) + (4j \times -6j) \][/tex]
Calculate each term separately:
- [tex]\( 3 \times 4 = 12 \)[/tex]
- [tex]\( 3 \times -6j = -18j \)[/tex]
- [tex]\( 4j \times 4 = 16j \)[/tex]
- [tex]\( 4j \times -6j = -24j^2 \)[/tex]
- Since [tex]\( j^2 = -1 \)[/tex], [tex]\( -24j^2 = -24(-1) = 24 \)[/tex]
Now combine these results:
[tex]\[ 12 - 18j + 16j + 24 \][/tex]
Simplify by combining like terms:
- The real parts: [tex]\( 12 + 24 = 36 \)[/tex]
- The imaginary parts: [tex]\( -18j + 16j = -2j \)[/tex]
Thus, the voltage is:
[tex]\[ 36 - 2j \][/tex]
Referencing the options provided:
[tex]\[ 36 - 2j \text{ volts} \][/tex]
Therefore, the correct answer is:
[tex]\[ 36 - 2j \text{ volts} \][/tex]
You should choose the option:
[tex]\[ \boxed{36 - j 2 \text{ volts}} \][/tex]
