Problem 55 - College Algebra Essentials

Question

Complex numbers are used in electronics to describe the current in an electric circuit. Ohm's law relates the current in a circuit, \(I\), in amperes, the voltage of the circuit, \(E,\) in volts, and the resistance of the circuit, \(R,\) in ohms, by the formula \(E=I R .\) Use this formula to solve Exercises \(55-56\) Find \(E,\) the voltage of a circuit, if \(I=(4-5 i)\) amperes and \(R=(3+7 i)\) ohms.

Step-by-Step Solution

Answer

The voltage (\(E\)) of the circuit is \(47 + 13i\) volts.

Step 1: Identify given quantities

The current (\(I\)) in the circuit is given as \(4 - 5i\) amperes and the resistance (\(R\)) of the circuit is \(3 + 7i\) ohms.

Step 2: Use Ohm's law to find voltage

Use Ohm's law, \(E=IR\), the voltage (\(E\)) in a circuit can be found by multiplying the current (\(I\)) by the resistance (\(R\)). Therefore, \(E=(4-5i)(3+7i)\).

Step 3: Compute the multiplication

We have to remember the formula \(i^2 = -1\) when expanding the brackets: \(E= 12 + 28i -15i -35i^2 = 12 + 13i + 35 = 47 + 13i\).

Key Concepts

Complex Numbers

Complex numbers are fundamental in various scientific and engineering disciplines, including electrical engineering where they are used to represent electrical properties like current, voltage, and resistance. A complex number is composed of two parts: a real part and an imaginary part, represented as a combination of a and bi, where a is the real part, b is the imaginary coefficient, and i is the imaginary unit with the property that i^2 = -1. This unique structure allows complex numbers to express oscillations, rotations, and periodic functions, which are prevalent in alternating current (AC) circuits.

Electrical Circuits

In the context of electrical circuits, Ohm's law is a fundamental principle stating that the voltage across a conductor is directly proportional to the current flowing through it, as long as the temperature remains constant. Ohm's law is represented by the equation E = IR, where E is the voltage, I is the current, and R is the resistance. When it comes to AC circuits or circuits with reactive components, the values of these variables can be complex numbers to account for the phase difference between the voltage and the current, as well as reactive components like inductors and capacitors.

Multiplication of Complex Numbers

When dealing with multiplication of complex numbers such as in Ohm's law applied to AC circuits, it's essential to understand how to correctly multiply these numbers. If we have two complex numbers, (a + bi) and (c + di), their product is found by multiplying both numbers in a distributive manner while remembering that i^2 = -1. The product is (ac - bd) + (ad + bc)i. For instance, if we multiply (4 - 5i) and (3 + 7i) as in our exercise, we will distribute and combine like terms accordingly, factoring in the value of i^2 to obtain the final complex voltage.