Problem 13 - Technical Mathematics with Calculus

Question

Convert the following metric units. Write your answer in scientific notation if the numerical value is greater than 1000 or less than 0.1. \(6.2 \times 10^{9}\) ohms to megohms.

Step-by-Step Solution

Answer

\(6.2 \times 10^{3}\) megohms

Step 1: Understand the Conversion

1 megohm (MΩ) is equal to 1,000,000 ohms. The task is to convert ohms to megohms.

Step 2: Set Up the Conversion

To convert ohms to megohms, divide the number of ohms by 1,000,000.

Step 3: Perform the Calculation

Divide 6.2 by 1,000,000 and retain the exponent to be in scientific notation. This yields: \(6.2 \times 10^{9} \div 1\times10^{6} = 6.2 \times 10^{(9-6)} = 6.2 \times 10^{3}\) megohms.

Step 4: Consider Scientific Notation Thresholds

Since the numerical value is greater than 1000, we convert it to scientific notation. Therefore, \(6.2 \times 10^{3}\) megohms is already in the correct scientific notation form and no further conversion is needed.

Key Concepts

Understanding Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It is commonly used by scientists, engineers, and mathematicians to make calculations simpler and to easily signify the number of significant figures.

In scientific notation, a number is written as the product of two parts: a decimal and an exponent. The decimal is typically a number between 1 and 10, and it is multiplied by 10 raised to an exponent. The exponent indicates how many places the decimal point moves to the right for positive exponents, or to the left for negative exponents. For example, the number 5000 can be written as 5 x 10^3 in scientific notation.

Measuring Electrical Resistance

Electrical resistance measurement is fundamental in the field of electronics and electrical engineering. Resistance, measured in ohms (Ω), quantifies how strongly a material opposes the flow of electric current. A higher resistance means less current flows for a given voltage, as per Ohm's Law, which states that current (I) is equal to voltage (V) divided by resistance (R), or I = V/R.

To measure resistance, multimeters and other specialized instruments are used. Resistance can vary widely depending on the material, thickness, length, and temperature, making the right measurement tools and techniques critical. Understanding these principles is important for diagnosing electrical issues, designing circuits, and ensuring the safe use of electrical devices.

Converting Megohms to Ohms

Conversion between different units of resistance, such as megohms (MΩ) to ohms (Ω), is an essential skill in electrical engineering. 1 megohm is equal to 1,000,000 ohms, reflecting a much higher resistance value. On a larger scale, such conversions are important for dealing with the varying resistance levels in electrical systems.

To convert from megohms to ohms, you need to multiply the value in megohms by one million. Conversely, to go from ohms to megohms, you divide by one million. For instance, a reading of 6.2 x 10^3 megohms is equivalent to simply 6.2 x 10^9 ohms, as you are moving the decimal point six places to the right (

Example Conversion

For 6.2 megohms, calculate 6.2 x 1,000,000, which gives us 6,200,000 ohms or 6.2 x 10^6 ohms in scientific notation). This process ensures that large resistance values can be more manageable and comparable.