Question

Three resistors, having resistances of \(4.98 \times 10^{5} \Omega, 2.47 \times 10^{4} \Omega,\) and \(\left.9.27 \times 10^{6} \Omega, \text { are wired in series (Fig. } 1-10\right) .\) Find the total resistance, using the formula \(R=R_{1}+R_{2}+R_{3}\)

Short answer

The total resistance is approximately 9.79 x 10^6 Ohm.

Step-by-step solution

Identify Resistances

Start by identifying the resistances of the three resistors which are given: Resistor 1, R1 = 4.98 x 10^5 Ohm, Resistor 2, R2 = 2.47 x 10^4 Ohm, and Resistor 3, R3 = 9.27 x 10^6 Ohm.

Apply Series Resistance Formula

To find the total resistance of resistors in series, use the formula Rtotal = R1 + R2 + R3. Simply add the values of R1, R2, and R3.

Calculate Total Resistance

Calculate the total resistance by adding the resistances together. Rtotal = (4.98 x 10^5) + (2.47 x 10^4) + (9.27 x 10^6) Ohm.

Convert to Standard Notation

Add the resistances making sure to align the decimal places correctly based on the exponent of 10. Rtotal = (498,000) + (24,700) + (9,270,000) Ohm.

Final Total Resistance

Add the resistances from the previous step to find the total resistance. Rtotal = 498,000 + 24,700 + 9,270,000 Ohm = 9,792,700 Ohm. Convert this back into scientific notation if necessary: Rtotal = 9.79 x 10^6 Ohm when rounded to three significant digits.

Key concepts

Electrical Resistance

Electrical resistance, often symbolized as 'R', is an essential concept in the world of electronics. It's a measure of the opposition to the flow of electric current in a conductor. The higher the resistance, the more difficult it is for the current to flow through a material. Resistors, the components that introduce this resistance into a circuit, come in various values to control the flow of current.

When resistors are connected in series, their total resistance is simply the sum of their individual resistances, because the current flows through each resistor one after the other. This series arrangement ensures that the current across each resistor remains the same, but the voltage across each one depends on its resistance.

Ohm's Law

Ohm's Law is one of the fundamental principles in the study of electricity. It's named after Georg Simon Ohm, who described the relationship between voltage (V), current (I), and resistance (R) in his famous equation: \[ V = I \times R \.\]
Ohm's Law tells us that the current flowing through a conductor between two points is directly proportional to the voltage across the two points, provided the temperature remains constant. It also implies that for a given voltage, the current is inversely proportional to the resistance. Understanding Ohm's Law is crucial for analyzing the behavior of electrical circuits and solving problems related to voltage, current, and resistance.

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It's written as the product of two factors: a coefficient (a number between 1 and 10) and a power of 10. For example, the number 2,000 can be expressed as \( 2 \times 10^3 \.\)

This form is particularly useful in circuit analysis and physics, as it allows us to work with very large or very small values, such as the resistance values in the exercise, efficiently. By using scientific notation, we can easily add or multiply these values without having to work with unwieldy long numbers. It's imperative to understand scientific notation to accurately perform calculations and convert between notation forms as needed.

Circuit Analysis

Circuit analysis is the process of determining the voltages across, and the currents through, every component in the electrical circuit. Proper analysis allows for the safe and efficient design and operation of electrical circuits. In the context of series circuits, such as the one in our exercise, circuit analysis involves determining the total resistance and anticipating how it affects the overall behavior of the circuit.

For instance, when resistors are connected in series, you must add their resistances to find the total resistance faced by the current. This sum impacts how much current flows as per Ohm's Law and influences how the voltage is distributed across each component. Mastering circuit analysis requires a good understanding of series and parallel circuits, Ohm's Law, and the ability to use scientific notation to deal with a variety of component values.