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Chapter 15: Problem 10

Write the permittivity \(10^{-5} \mathrm{~F} / \mathrm{m}\) using base units only, and without the use of a negative exponent or a decimal point.

Short Answer

Expert verified
Question: Convert the given permittivity value, \(10^{-5} \mathrm{~F} / \mathrm{m}\), into base units without using negative exponents or decimal points. Answer: \(\frac{\mathrm{kg^{-1}} \cdot \mathrm{m^{-1}} \cdot \mathrm{s^2} \cdot \mathrm{A^2}}{100000}\)

Step by step solution

01

Understanding permittivity units

Permittivity is usually measured in Farads per meter (F/m). The base units for the Farad are Coulombs per Volt (C/V), where Coulomb (C) is the unit for electric charge and Volt (V) is the unit for electric potential. The Volt can be further broken down into base SI units: Joules per Coulomb (J/C). And finally, a Joule (J) is a unit of energy and can be represented in terms of base SI units as \(\mathrm{kg} \cdot \mathrm{m^2} / \mathrm{s^2}\). So in the end, the base units for permittivity are \(\mathrm{kg} \cdot \mathrm{m^{-3}} \cdot \mathrm{s^4} \cdot \mathrm{A^2}\), where A is the unit for electric current (Ampere).
02

Convert the given value

Now, we have the permittivity value \(10^{-5} \mathrm{~F} / \mathrm{m}\). First, let's convert Farads to base SI units: \(10^{-5} \mathrm{~F} / \mathrm{m} = 10^{-5} \cdot (\mathrm{C} / \mathrm{V}) / \mathrm{m}\) \(= 10^{-5} \cdot (\mathrm{C} / (\mathrm{J} / \mathrm{C})) / \mathrm{m}\) Next, convert Joules and simplify the expression: \(= 10^{-5} \cdot (\mathrm{C} / (\mathrm{kg} \cdot \mathrm{m^2} / (\mathrm{s^2} \cdot \mathrm{C}))) / \mathrm{m}\) \(= 10^{-5} \cdot (\mathrm{kg^{-1}} \cdot \mathrm{m^{-1}} \cdot \mathrm{s^2} \cdot \mathrm{A^2})\)
03

Remove the negative exponent

Now, we need to rewrite the expression without a negative exponent. To do this, we'll move the \(10^{-5}\) term to the denominator: \(= \frac{1}{10^5} \cdot (\mathrm{kg^{-1}} \cdot \mathrm{m^{-1}} \cdot \mathrm{s^2} \cdot \mathrm{A^2})\)
04

Express the permittivity without a decimal point

To remove the use of the decimal point, we can rewrite the fraction as: \(= \frac{1}{100000} \cdot (\mathrm{kg^{-1}} \cdot \mathrm{m^{-1}} \cdot \mathrm{s^2} \cdot \mathrm{A^2})\)
05

Combine the values to obtain the final result

Finally, we'll combine the values to get the permittivity value in base units without a negative exponent or decimal point: \(= \frac{\mathrm{kg^{-1}} \cdot \mathrm{m^{-1}} \cdot \mathrm{s^2} \cdot \mathrm{A^2}}{100000}\)

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