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Chapter 4: Problem 36

Approximate the change in the magnitude of the electrostatic force between two charges when the distance between them increases from \(r=20 \mathrm{m}\) to \(r=21 \mathrm{m}\left(F(r)=0.01 / r^{2}\right)\)

Short Answer

Expert verified
Answer: The approximate change in magnitude of the electrostatic force between the two charges is approximately \(-2.324 \times 10^{-6} N\) when the distance increases from 20 m to 21 m.

Step by step solution

01

Find the Force at Initial Distance

Using the given function, we will first find the force at the initial distance (\(r = 20m\)). Substitute \(r = 20\) in the given function \(F(r) = \frac{0.01}{r^2}\): \(F(20) = \frac{0.01}{(20)^2}\)
02

Simplify Expression for Initial Force

Simplify the expression obtained in step 1: \(F(20) = \frac{0.01}{400}\)
03

Calculate Initial Force

Now, evaluate the value of the initial force: \(F(20) = 2.5 \times 10^{-5} N\)
04

Find the Force at Final Distance

Next, find the force at the final distance (\(r = 21m\)). Substitute \(r = 21\) in the given function \(F(r) = \frac{0.01}{r^2}\): \(F(21) = \frac{0.01}{(21)^2}\)
05

Simplify Expression for Final Force

Simplify the expression obtained in step 4: \(F(21) = \frac{0.01}{441}\)
06

Calculate Final Force

Now, evaluate the value of the final force: \(F(21) = 2.2675737 \times 10^{-5} N\)
07

Find the Change in Force

To find the change in force, we need to find the difference between the magnitudes of the force at both distances. Subtract the initial force value from the final force value: Change in Force = \(F(21) - F(20)\)
08

Calculate the Change in Force

Finally, calculate the value of the change in force: Change in Force = \((2.2675737 \times 10^{-5} N) - (2.5 \times 10^{-5} N)\) Change in Force = \(-2.324263 \times 10^{-6} N\) Thus, the approximate change in magnitude of the electrostatic force between the two charges is \(-2.324 \times 10^{-6} N\) when the distance increases from 20 m to 21 m.

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