Problem 81 - Precalculus Student Solutions Manual 5th

Question

Complex numbers are used to describe current I, voltage \(E,\) and impedance \(Z\) (the opposition to current). These three quantities are related by the equation \(E=I Z, \quad\) which is known as Ohm's Law. Thus, if any two of these quantities are known, the third can be found. In each exercise, solve the equation \(E=I Z\) for the remaining value. $$I=5+7 i, Z=6+4 i$$

Step-by-Step Solution

Answer

The voltage (E) is \( 2 + 62i \).

Step 1: Identify the given quantities

Identify the values of current (I) and impedance (Z) from the problem statement. Here, current is given as \( I = 5 + 7i \) and impedance is given as \( Z = 6 + 4i \).

Step 2: Recall Ohm's Law

Recall the formula relating voltage (E), current (I), and impedance (Z), which is given by Ohm's Law: \( E = I Z \).

Step 3: Multiply the complex numbers

Multiply the given values of current and impedance. Use the distributive property and the fact that \( i^2 = -1 \): \( (5 + 7i)(6 + 4i) = 5 \times 6 + 5 \times 4i + 7i \times 6 + 7i \times 4i \).

Step 4: Simplify the expression

Perform the multiplications: \( 5 \times 6 = 30 \), \( 5 \times 4i = 20i \), \( 7i \times 6 = 42i \), \( 7i \times 4i = 28i^2 \).

Step 5: Combine like terms

Combine the real parts and the imaginary parts: \( 30 + 20i + 42i + 28i^2 \).

Step 6: Substitute \( i^2 = -1 \)

Since \( i^2 = -1 \), replace \( 28i^2 \) with \( 28(-1) = -28 \): \( 30 + 20i + 42i - 28 \).

Step 7: Simplify final expression

Combine the real parts and the imaginary parts: \( (30 - 28) + (20i + 42i) = 2 + 62i \).

Step 8: Write the solution

The voltage (E) is thus \( E = 2 + 62i \).

Key Concepts

complex numbers

Complex numbers are essential in electrical engineering to describe quantities such as current, voltage, and impedance. A complex number is in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) represents the square root of -1. Understanding how to handle complex numbers is crucial when dealing with alternating current (AC) circuits.

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering. It states that the voltage \(E\) across a conductor is proportional to the current \(I\) flowing through it and the impedance \(Z\) of the conductor: \(E = IZ\). In AC circuits, Ohm's Law extends to complex numbers to accommodate the phase differences between voltage and current. By knowing any two of the quantities \(E\), \(I\), or \(Z\), one can calculate the third using this simple relationship.

electrical impedance

Electrical impedance \(Z\) quantifies the opposition that a circuit presents to the flow of alternating current. It is a complex number that combines resistance (which affects both DC and AC) and reactance (which affects only AC). The impedance can be expressed as \(Z = R + jX\), where \(R\) is the resistance and \(X\) is the reactance, and \(j\) is the imaginary unit equivalent to \(i\) in electrical engineering notation. Effective handling of impedance is crucial for analyzing AC circuits and their behavior.

current and voltage relationship

In any electrical circuit, there is an inherent relationship between current and voltage. Using Ohm's Law, this relationship is expressed as: \(E = IZ\). This means if you know the current \(I\) and impedance \(Z\), you can find the voltage \(E\). For instance, in our exercise, the current is \(I = 5 + 7i\) and the impedance \(Z = 6 + 4i\). Multiplying these complex numbers gives us the voltage \(E = (5 + 7i)(6 + 4i) = 2 + 62i\). This relationship is key to solving many problems in AC circuits.

complex multiplication

Complex multiplication involves expanding the product of two complex numbers as you would with polynomials, remembering that \(i^2 = -1\). For example, to find \((5 + 7i)(6 + 4i)\), you distribute each term: \(5 \times 6 + 5 \times 4i + 7i \times 6 + 7i \times 4i\). This becomes \(30 + 20i + 42i + 28i^2\). Since \(i^2 = -1\), \(28i^2\) translates to \(-28\). Finally, combine like terms to simplify: \(30 - 28 + 62i = 2 + 62i\). Mastery of complex multiplication is essential for working with AC circuits and impedance calculations.