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Chapter 5: Problem 13

(a) What is the electrostatic potential energy (in joules) between an electron and a proton that are separated by \(230 \mathrm{pm}\) ? (b) What is the change in potential energy if the distance separating the electron and proton is increased to \(1.0 \mathrm{nm}\) ? (c) Does the potential energy of the two particles increase or decrease when the distance is increased to \(1.0 \mathrm{nm}\) ?

Short Answer

Expert verified
a) The electrostatic potential energy between the electron and proton separated by 230 pm is: \(U_1 = \dfrac{(8.99 * 10^9 N m^2/C^2)(1.60 * 10^{-19} C)(-1.60 * 10^{-19} C)}{230 * 10^{-12} m} \approx -4.99 * 10^{-18} J\) b) The electrostatic potential energy between the electron and proton separated by 1.0 nm is: \(U_2 = \dfrac{(8.99 * 10^9 N m^2/C^2)(1.60 * 10^{-19} C)(-1.60 * 10^{-19} C)}{1.0 * 10^{-9} m} \approx -2.30 * 10^{-18} J\) c) The change in potential energy is: \(\Delta U = U_2 - U_1 \approx -2.30 * 10^{-18} J - (-4.99 * 10^{-18} J) \approx 2.69 * 10^{-18} J\) Since \(\Delta U > 0\), the potential energy of the two particles increases when the distance is increased to 1.0 nm.

Step by step solution

01

Convert Distances to Meters

First, we need to convert the given distances to meters to be compatible with the units of the electrostatic constant: - 230 pm = 230 * 10^(-12) meters - 1.0 nm = 1.0 * 10^(-9) meters
02

Constants and Initial Values

For solving this problem, we require the following constants and initial values: - \(k_e = 8.99 * 10^9 \ N m^2/C^2\) (electrostatic constant) - Charge of a proton, \(q_p = 1.60 * 10^{-19} C\) - Charge of an electron, \(q_e = -1.60 * 10^{-19} C\)
03

Calculate Potential Energy at 230 pm

Using the electrostatic potential energy formula for a distance of 230 pm (230 * 10^(-12) meters): \(U_1 = \dfrac{(8.99 * 10^9 N m^2/C^2)(1.60 * 10^{-19} C)(-1.60 * 10^{-19} C)}{230 * 10^{-12} m}\) Calculate \(U_1\) to find the potential energy.
04

Calculate Potential Energy at 1.0 nm

Using the electrostatic potential energy formula for a distance of 1.0 nm (1.0 * 10^(-9) meters): \(U_2 = \dfrac{(8.99 * 10^9 N m^2/C^2)(1.60 * 10^{-19} C)(-1.60 * 10^{-19} C)}{1.0 * 10^{-9} m}\) Calculate \(U_2\) to find the potential energy.
05

Change in Potential Energy

To find the change in potential energy, subtract the initial potential energy from the final potential energy: \(\Delta U = U_2 - U_1\) Calculate \(\Delta U\) to find the change in potential energy.
06

Potential Energy Increase or Decrease

Determine if the potential energy increased or decreased by analyzing the sign of \(\Delta U\). If \(\Delta U > 0\), the potential energy increased; if \(\Delta U < 0\), the potential energy decreased.

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