Question

(a) What is the electrostatic potential energy (in joules) between an electron and a proton that are separated by 53 pm? (b) What is the change in potential energy if the distance separating the electron and proton is increased to 1.0 nm? (c) Does the potential energy of the two particles increase or decrease when the distance is increased to 1.0 nm?

Short answer

(a) The initial electrostatic potential energy (U_initial) between an electron and a proton separated by 53 pm is calculated as: $U_{initial}=(8.99\times {10}^{9}N{m}^2{C}^{-2})\times (-1.6\times {10}^{-19}C)\times (1.6\times {10}^{-19}C)/(53\times {10}^{-12}m)$. (b) When the distance increases to 1.0 nm, the final electrostatic potential energy (U_final) is: $U_{final}=(8.99\times {10}^{9}N{m}^2{C}^{-2})\times (-1.6\times {10}^{-19}C)\times (1.6\times {10}^{-19}C)/(1.0\times {10}^{-9}m)$. (c) To find the change in potential energy, we calculate the difference $\Delta U=U_{final}-U_{initial}$. If ΔU is positive, the potential energy increases, and if ΔU is negative, the potential energy decreases.

Step-by-step solution

a) Calculate initial electrostatic potential energy

First, we will determine the initial electrostatic potential energy when the electron and proton are separated by 53 pm. We know that the charge of an electron (q1) is -1.6 × 10^-19 C, the charge of a proton (q2) is +1.6 × 10^-19 C, and the initial distance (r) is 53 pm (which we need to convert to meters: 53 × 10^-12 m). Using the formula U = k × (q1 × q2) / r, we can calculate the initial electrostatic potential energy: U_initial = (8.99 × 10^9 N m^2 C^-2) × (-1.6 × 10^-19 C × 1.6 × 10^-19 C) / (53 × 10^-12 m)

b) Calculate final electrostatic potential energy

Now we will calculate the final electrostatic potential energy when the electron and proton are separated by 1.0 nm (which we also need to convert to meters: 1.0 × 10^-9 m). Their charges remain the same, so we can use the same formula: U_final = (8.99 × 10^9 N m^2 C^-2) × (-1.6 × 10^-19 C × 1.6 × 10^-19 C) / (1.0 × 10^-9 m)

c) Calculate the change in potential energy

To find the change in potential energy between the initial and final situations, we can subtract the initial potential energy from the final potential energy: ΔU = U_final - U_initial

d) Determine if the potential energy increases or decreases

Finally, we will determine whether the potential energy increases or decreases when the distance between the electron and proton increases from 53 pm to 1.0 nm. If ΔU is positive, the potential energy increases; if ΔU is negative, the potential energy decreases.

Key concepts

Charge Interaction

In the realm of electrostatics, understanding charge interaction is fundamental. Charges come in two types: positive and negative. They interact with each other through electric forces. But how exactly do these charges interact?
In simple terms:

• Opposite charges (positive and negative) attract each other.
• Like charges (positive-positive or negative-negative) repel each other.

This pull or push between charges forms the basis of electrostatic potential energy. The potential energy depends directly on the magnitude of the interacting charges. For instance, in this exercise, we have an electron and a proton. The electron has a negative charge, while the proton has a positive charge. Due to their opposite charges, they are naturally attracted to each other, which lowers the system's energy.
The formula for calculating electrostatic force between two point charges is given by Coulomb's Law: $$F=k\times \frac{|q_1\times q_2|}{r^2}$$where:- $F$ is the electrostatic force,- $q_1$ and $q_2$ are the charges,- $r$ is the distance between them,- $k$ is Coulomb's constant, approximately $8.99\times {10}^9{\text{N m}}^2{\text{C}}^{-2}$.Understanding this interaction is crucial, as it plays a significant role in determining the potential energy of a system.

Distance Dependence

Distance is another crucial factor in electrostatic potential energy. It dictates not only the magnitude of force between charges but also the energy within the system.
Specifically, as the distance between two charges changes, the electrostatic potential energy follows this pattern:

• As distance decreases, potential energy increases (making the system more unstable).
• As distance increases, potential energy decreases (stabilizing the system).

For our problem, when an electron and a proton are separated by a distance of 53 pm, they have a specific potential energy based on that proximity. Using the formula:$$U=k\times \frac{q_1\times q_2}{r}$$where $U$ represents the potential energy, we can calculate how energy evolves as the distance between the charges changes. When the distance increases to 1.0 nm, the reduction in potential energy signifies that the charges are now less strongly interacting, signaling a more stable state. Remember that both distance and energy are inversely proportional: doubling the distance doesn’t double the potential energy but rather halves it.

Potential Energy Change

Potential energy change is another vital aspect that gives insight into how systems adjust when configurations alter. When you change the distance between charges, you inherently change the amount of potential energy in the system.
Let's delve deeper into potential energy change:

• Determine both initial and final potential energies using the previously mentioned formula.
• Compute the change in potential energy: $$\mathrm{\Delta }U=U_{\text{final}}-U_{\text{initial}}$$

To conclude, understanding whether the potential energy increases or decreases is key. Interestingly, in our scenario where distance increases, potential energy decreases. This decrease is confirmed by observing a negative value in $\mathrm{\Delta }U$. It means the system loses potential energy as the charges become less effective due to increased separation. This context shows a shift towards stability as an electron and proton find themselves further apart, thereby reducing the electrostatic attraction between them.