Problem 55 - Algebra and Trigonometry

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 \(E\) across the circuit is \(E=-23+13i\) volts.

Step 1: Understand the Problem

Given that the current \(I=(4-5i)\) amperes and the resistance \(R=(3+7i)\) ohms in a circuit. The task is to find the voltage \(E\) in volts. By Ohm's law, \(E=I \cdot R\), which in this is the multiplication of two complex numbers.

Step 2: Multiply the Complex Numbers

To find the product of the two complex numbers, apply the distributive law:\n\[E=(4-5i)(3+7i) = 12+28i-15i-35 = -23+13i\]

Step 3: Interpret the Result

The voltage \(E=-23+13i\) volts. The real part is the resistive or in-phase voltage, and the imaginary part is the reactive or quadrature voltage in the circuit. Thus, the voltage across the circuit is \(E=-23+13i\) volts.

Key Concepts

Ohm's law

Ohm's Law is a fundamental principle used in electrical engineering to describe the relationships between voltage, current, and resistance in electric circuits. It's expressed with the simple equation:\[ E = I \cdot R \] where:- **\(E\)** is the voltage measured in volts.- **\(I\)** is the current measured in amperes (amps).- **\(R\)** is the resistance measured in ohms.Ohm's Law helps us understand how electricity behaves in circuits. By knowing two of the three variables, you can easily calculate the third. In our exercise, the current and resistance are given as complex numbers. This adds a layer of complexity, requiring the use of complex multiplication to find the voltage. Remember, Ohm's Law applies directly in circuits without frequency variations or reactive components. In those scenarios, the interactions between current and voltage become phase-dependent, which is why complex numbers often come into play.

Complex multiplication

Complex multiplication is a process where two complex numbers are multiplied together. Each complex number is in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. When multiplying two complex numbers, we use the distributive property:For given numbers \((4-5i)\) and \((3+7i)\), the multiplication would be:1. Multiply the real parts: \(4 \times 3 = 12\).2. Multiply the real part of one by the imaginary part of the other: \(4 \times 7i = 28i\) and \(-5i \times 3 = -15i\).3. Multiply the imaginary parts: \(-5i \times 7i = -35i^2\). Here, \(i^2 = -1\), so \(-35i^2 = 35\).Adding these together, the complex product is:\[ 12 + 28i - 15i + 35 = -23 + 13i \]Each step involves keeping track of the real and imaginary components. Complex multiplication results in a new complex number that effectively combines both numbers' magnitudes and angles.

Electric circuits

Electric circuits are pathways through which electricity flows, typically consisting of a closed loop made up of several components like resistors, capacitors, and power sources. In its simplest form, a circuit might include a power source (like a battery) and a load (like a light bulb), joined by conductive wires. The current moves from the positive side of the power source, passes through the circuit, and returns to the negative side. In more complex circuits, like those with alternating current (AC), complex numbers are used to describe aspects like impedance, which combines resistance (real part) and reactance (imaginary part). The imaginary part signifies components like capacitors and inductors, which introduce a phase shift between voltage and current. When analyzing these circuits, engineers need to consider the frequencies at which the circuits operate, as different components will react differently at varying frequencies. Complex numbers provide a convenient way to handle these phase shifts and help engineers design circuits that perform efficiently under specified conditions. Ultimately, understanding the role of complex numbers in electric circuits allows for advanced electrical system designs, ensuring the proper functioning of everything from household appliances to complex industrial equipment.