Problem 83 - 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. $$I=10+4 i, E=88+128 i$$

Step-by-Step Solution

Answer

Z = 12 + 8i

Step 1: Write down Ohm's Law

The relationship between current, voltage, and impedance is given by Ohm's Law: \[ E = IZ \]

Step 2: Identify the given quantities and what needs to be found

Given values are: \( I = 10 + 4i \) \( E = 88 + 128i \) We need to find the impedance \( Z \).

Step 3: Rearrange Ohm's Law to solve for Z

Rearrange the equation \[ E = IZ \] to solve for \( Z \): \[ Z = \frac{E}{I} \]

Step 4: Substitute the known values into the equation

Substitute \( E = 88 + 128i \) and \( I = 10 + 4i \) into the rearranged equation: \[ Z = \frac{88 + 128i}{10 + 4i} \]

Step 5: Perform the division of complex numbers

To divide the complex numbers, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \( 10 + 4i \) is \( 10 - 4i \). So: \[ Z = \frac{(88 + 128i)(10 - 4i)}{(10 + 4i)(10 - 4i)} \]

Step 6: Simplify the denominator

Simplify the denominator using the formula \( (a + bi)(a - bi) = a^2 - b^2i^2 \): \[ (10 + 4i)(10 - 4i) = 10^2 - (4i)^2 = 100 - 16(-1) = 100 + 16 = 116 \]

Step 7: Expand the numerator

Expand the numerator: \[ (88 + 128i)(10 - 4i) = 88*10 + 88*(-4i) + 128i*10 + 128i*(-4i) \] \[ = 880 - 352i + 1280i - 512i^2 \] Since \( i^2 = -1 \): \[ = 880 - 352i + 1280i + 512 \] Combine like terms: \[ = 1392 + 928i \]

Step 8: Divide the simplified numerator by the simplified denominator

Divide the simplified numerator by the simplified denominator: \[ Z = \frac{1392 + 928i}{116} \] Perform the division for both real and imaginary parts: \[ Z = \frac{1392}{116} + \frac{928i}{116} \] \[ Z = 12 + 8i \]

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle used in electrical engineering and physics, describing the relationship between voltage (E), current (I), and impedance (Z
). The law is represented by the equation: E = I Z.
This means that the voltage across a component is equal to the current flowing through it multiplied by its impedance.

In this context:

• E: Voltage, measured in volts (V)


• I: Current, measured in amperes or amps (A)


• Z: Impedance, measured in ohms (Ω)

This law allows us to calculate one of the three quantities if we know the other two. For example, if we know the current and the impedance, we can find the voltage by multiplying them together.
Understanding Ohm's Law is essential for solving problems involving complex numbers in electrical circuits.

Complex Number Division

Dividing complex numbers can seem tricky, but it follows a straightforward process. Complex numbers have a real part and an imaginary part, typically represented as a + bi .
When we divide complex numbers, we use the concept of the conjugate to simplify the division.

In our example, we need to find the impedance (Z) by dividing two complex numbers (88 + 128i) by (10 + 4i).
Here's how you do it step-by-step:

• Multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of 10 + 4i is 10 - 4i.


• Expand the numerator: (88 + 128i)(10 - 4i).


• Simplify the denominator using the formula for the product of a complex number and its conjugate: (a + bi)(a - bi) = a^2 - b^2
  
• Simplify the resulting expressions to get a final real and imaginary component for the quotient.

This process ensures that you correctly perform the division and convert the result back into a standard complex number form.

Impedance

Impedance (Z) is a measure of how much a circuit resists the flow of alternating current (AC).
It is similar to resistance in direct current (DC) circuits but also takes into account reactance, which arises from capacitors and inductors in the circuit and is frequency-dependent.

Impedance is a complex quantity, meaning it has both a real part (resistance R) and an imaginary part (reactance X). It is represented as:

• Z = R + Xi

Reactance can be further divided into:

• Capacitive reactance (X_C), which decreases as frequency increases.


• Inductive reactance (X_L), which increases as frequency increases.

Understanding impedance is crucial because it allows engineers to design circuits that work correctly at different frequencies. For instance, in our problem, knowing the impedance helps us understand how the circuit impedes the current flow, allowing us to adjust components to achieve desired electrical characteristics.