Problem 2 - College Physics

Question

(a) How many ohms are there in a 7.85 -megohm resistor? (b) Typical laboratory capacitors are around 5 picofarads. How many farads are they? (c) The speed of light in vacuum is \(3.00 \times 10^{8} \mathrm{m} / \mathrm{s} .\) Express this speed in gigameters per second. (d) The wavelength of visible light is between 400 \(\mathrm{nm}\) and 700 \(\mathrm{nm} .\) Express this wavelength in meters. (e) The diameter of a typical atomic nucleus is about 2 femtometers. Express this diameter in meters.

Step-by-Step Solution

Answer

(a) 7,850,000 ohms (b) 5 × 10^{-12} farads (c) 0.3 gigameters/second (d) 4.00 × 10^{-7} to 7.00 × 10^{-7} meters (e) 2 × 10^{-15} meters.

Step 1: Convert Megohms to Ohms

To convert from megohms to ohms, remember that 1 megohm is equal to \( 10^6 \) ohms. Therefore, to convert 7.85 megohms to ohms, multiply 7.85 by \( 10^6 \): \[ 7.85 \times 10^6 \text{ ohms} = 7,850,000 \text{ ohms}. \]

Step 2: Convert Picofarads to Farads

To convert from picofarads to farads, note that 1 picofarad is equal to \( 10^{-12} \) farads. Therefore, 5 picofarads can be converted to farads by multiplying by \( 10^{-12} \): \[ 5 \times 10^{-12} \text{ farads} = 5 \times 10^{-12} \text{ farads}. \]

Step 3: Convert Meters per Second to Gigameters per Second

The speed of light is given as \( 3.00 \times 10^8 \text{ meters per second} \). Since 1 gigameter is equal to \( 10^9 \) meters, to convert to gigameters per second, divide by \( 10^9 \): \[ \frac{3.00 \times 10^8}{10^9} \text{ gigameters per second} = 0.3 \text{ gigameters per second}. \]

Step 4: Convert Nanometers to Meters

To convert the wavelength range from nanometers to meters, use the conversion factor that 1 nanometer is \( 10^{-9} \) meters. Therefore, the range 400 nm to 700 nm converts as follows: \[ 400 \times 10^{-9} \text{ meters} = 4.00 \times 10^{-7} \text{ meters} \] and \[ 700 \times 10^{-9} \text{ meters} = 7.00 \times 10^{-7} \text{ meters}. \]

Step 5: Convert Femtometers to Meters

To convert the diameter of an atomic nucleus from femtometers to meters, note that 1 femtometer is equal to \( 10^{-15} \) meters. Thus, for a diameter of 2 femtometers: \[ 2 \times 10^{-15} \text{ meters}. \]

Key Concepts

Megohms to Ohms

When you're converting megohms to ohms, you're working with a simple unit conversion that involves understanding the power of ten relationships in electrical resistance. One megohm, abbreviated as MΩ, is equivalent to one million ohms. In scientific notation, this is represented as \(10^6\) ohms. This means you're essentially multiplying by a million to go from megohms to ohms.

Here's how you can remember it:

• "Mega" stands for million, so you're dealing with powers of 6 (in base 10).
• Therefore, to convert 7.85 MΩ to ohms, you perform the calculation \(7.85 \times 10^6\).
• The result is \(7,850,000\) ohms. This large number reflects the large capacity for electrical resistance when expressed in ohms.

This conversion is common in physics and engineering, especially when dealing with high-resistance components.

Picofarads to Farads

Converting picofarads to farads involves working with very small units of capacitance since "pico" refers to a trillionth (or \(10^{-12}\)) of a unit. Picofarads (pF) are often used in electronics where tiny charge capacities are common.

Understanding the conversion:

• A picofarad is a unit of capacitance equal to \(10^{-12}\) farads.
• Therefore, to convert 5 picofarads to farads, you multiply by \(10^{-12}\).
• The calculation yields \(5 \times 10^{-12}\) farads, which is an extremely small number, indicative of micro-level capacitance.

In laboratories and electronics, capacitors with such small values are used in high-frequency applications and sensitive circuits.

Nanometers to Meters

Nanometers are often used to measure things at an atomic or molecular scale, which makes understanding nanometer-to-meter conversions essential in both physics and chemistry. One nanometer is one-billionth of a meter, or \(10^{-9}\) meters.

How to do the conversion:

• The size of visible light wavelengths usually fall into the nanometer range, from 400 nm to 700 nm.
• To convert these numbers, multiply them by \(10^{-9}\).
• Thus, 400 nm becomes \(4.00 \times 10^{-7}\) meters, and 700 nm becomes \(7.00 \times 10^{-7}\) meters.

These scales are useful in fields such as optics and nanotechnology, where precision is key.

Speed of Light Conversion

The speed of light in a vacuum is famously known as approximately \(3.00 \times 10^8\) meters per second, a fundamental constant in physics. When converting this to gigameters per second, you're scaling it down to a larger unit because "giga" refers to a billion, or \(10^9\).

Steps to convert:

• 1 gigameter is equal to \(10^9\) meters.
• Therefore, to convert meters to gigameters, divide by \(10^9\).
• Thus, \(3.00 \times 10^8\) meters per second becomes \(0.3\) gigameters per second.

This scale offers a perspective suitable for astronomical distances, where expressing the speed of light in gigameters offers a more convenient, comprehensible figure.