Question

Two positive point charges of $12\mu \mathrm{c}$ and $8\mu \mathrm{c}$ are $10\mathrm{cm}$ apart each other. The work done in bringing them $4\mathrm{cm}$ closer is .... (A) $5.8\mathrm{J}$ (B) $13\mathrm{eV}$ (C) $5.8\mathrm{eV}$ (D) $13\mathrm{J}$

Short answer

The work done in bringing the two point charges 4 cm closer is 5.8 J or 3.62x10¹⁹ eV.

Step-by-step solution

List the given information

- Charge 1: $q_1=12\mu C$ - Charge 2: $q_2=8\mu C$ - Initial distance: $r_1=10cm=0.1m$ - Decrease in distance: $dr=4cm=0.04m$ - Final distance: $r_2=r_1-dr=0.1-0.04=0.06m$

Calculate the initial and final electric potential energy

Using the formula for electric potential energy between two point charges, we have: $U=\frac{k\cdot q_1\cdot q_2}{r}$ where $k=8.9875\times {10}^9\frac{N{m}^2}{C^2}$ is the electrostatic constant. We need to calculate the initial and final potential energies and then find the difference to determine the work done. Initial potential energy, $U_1$: $U_1=\frac{k\cdot q_1\cdot q_2}{r_1}$ Final potential energy, $U_2$: $U_2=\frac{k\cdot q_1\cdot q_2}{r_2}$

Calculate the work done

To calculate the work done, we find the difference between the initial and final potential energies: Work done ($W$) = $U_2-U_1$

Convert the work done to Joules and electron volts

We can now calculate the work done and also convert it to electron volts, using the conversion factor: $1eV=1.602\times {10}^{-19}J$ $W=U_2-U_1$ in Joules $W_{eV}=\frac{W}{1.602\times {10}^{-19}}$ in electron volts Now substitute the given values and calculate the work done to choose the correct option from the given choices.

Key concepts

Point Charges

Point charges are objects with a negligible size that have an electric charge. They can repel or attract each other based on their charge signs. Positive charges repel other positive charges and attract negative charges.
In our exercise, we have two positive point charges: one with a charge of $12\mu \mathrm{C}$ and another with $8\mu \mathrm{C}$.
When point charges interact, they create an electric field around them. The field is responsible for the force and potential energy between them.
This interaction is significant as it governs how charged particles behave in each other's presence.
Calculating electric potential energy involves these charges and the distance between them, which directly affects the force exerted by each on the other.

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states the force magnitude is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
The formula is: $F=\frac{k\cdot q_1\cdot q_2}{r^2}$, where:

• $F$ is the force between charges
• $k$ is Coulomb's constant, $8.9875\times {10}^9\frac{N{m}^2}{C^2}$
• $q_1$ and $q_2$ are the charges
• $r$ is the distance between the charges

In the context of potential energy, we simplify to $U=\frac{k\cdot q_1\cdot q_2}{r}$ since the energy is stored in their relative position, not their force interactions directly.
Understanding Coulomb's Law is crucial when calculating these energies, as it connects the micro-level interactions of charges to macroscopic forces and energies we can measure easily.

Work-Energy Principle

The work-energy principle is fundamental in physics. It explains how the work done on an object contributes to its energy. Specifically, in electric systems, the work done changes the electric potential energy.
If you move two point charges closer, work is done against the electric repulsive force, increasing potential energy. Conversely, moving them apart decreases potential energy.
The formula for work done when moving point charges is linked to changes in potential energy: $W=U_2-U_1$, where:

• $W$ is work done
• $U_1$ is initial potential energy
• $U_2$ is final potential energy

In our specific exercise, by bringing the charges 4 cm closer, we calculate and compare these energies to find the work done.
This principle helps us understand how energy changes in electric fields, making it a cornerstone in analyzing electric potential scenarios.

Electron Volt Conversion

Electron volts (eV) are a convenient unit of energy particularly in atomic and molecular scales. It's used when dealing with energies on the scale of electrons.
One electron volt is defined as the amount of kinetic energy gained by an electron when accelerated through a potential difference of one volt.
Conversion between joules and electron volts is essential in these problems because calculations initially give results in joules: $$1\mathrm{eV}=1.602\times {10}^{-19}\mathrm{J}$$By converting from joules to electron volts, we express energy in a more intuitive scale, particularly handy for practical applications involving small charge systems.
In the exercise, after finding work done in joules, we convert it to electron volts.This method of converting is essential to understanding and communicating results in scientific and engineering contexts.