Question

Two point charges \(100 \mu \mathrm{c}\) and \(5 \mu \mathrm{c}\) are placed at points \(\mathrm{A}\) and \(B\) respectively with \(A B=40 \mathrm{~cm}\). The work done by external force in displacing the charge \(5 \mu \mathrm{c}\) from \(\mathrm{B}\) to \(\mathrm{C}\) where \(\mathrm{BC}=30 \mathrm{~cm}\), angle \(\mathrm{ABC}=(\pi / 2)\) and \(\left[1 /\left(4 \pi \epsilon_{0}\right)\right]\) \(=9 \times 10^{9} \mathrm{Nm}^{2} / \mathrm{c}^{2}\). (A) \(9 \mathrm{~J}\) (B) \((9 / 25) \mathrm{J}\) (C) \((81 / 20) \mathrm{J}\) (D) \(-(9 / 4) \mathrm{J}\)

Short answer

The work done by the external force on the charge is \(\frac{9}{20}J\) which is equivalent to (C) \((81 / 20) \mathrm{J}\).

Step-by-step solution

Calculate initial electric potential energy

To find the initial electric potential energy, we will use the given values of charges and the distance between them. We have \(q_1 = 100\mu C\), \(q_2 = 5\mu C\) and \(r_{initial} = AB = 40cm\). Using the formula for electric potential energy: \(U_{initial} = \frac{1}{4\pi\epsilon_0}\cdot\frac{q_1q_2}{r_{initial}}\) \(U_{initial} = 9\times 10^9\mathrm{Nm^2/c^2}\cdot\frac{100\times 10^{-6}C\cdot 5\times 10^{-6}C}{0.4m}\)

Calculate the final electric potential energy

Now we have to find the electric potential energy at the final position. In order to find this, we first need to find the final distance between the charges. We are given that angle ABC = \(\pi/2\) and BC = 30cm. We can use the Pythagorean theorem to find the distance between charges A and C. \(AC = \sqrt{(AB)^2 + (BC)^2} = \sqrt{ (40cm)^2 + (30cm)^2 }\) Now we can calculate the final electric potential energy: \(U_{final} = \frac{1}{4\pi\epsilon_0}\cdot\frac{q_1q_2}{r_{final}}\) \(U_{final} = 9\times 10^9\mathrm{Nm^2/c^2}\cdot\frac{100\times 10^{-6}C\cdot 5\times 10^{-6}C}{AC}\)

Calculate the work done by external force

With the initial and final electric potential energies, we can now find the work done by the external force: \(W = U_{final}-U_{initial}\) \(W = 9\times 10^9\mathrm{Nm^2/c^2} \cdot 5\times 10^{-6}C\cdot 100\times 10^{-6}C \left( \frac{1}{AC} - \frac{1}{0.4m} \right)\) Now, substituting the value for AC: \(AC = \sqrt{ (40cm)^2 + (30cm)^2 } = \sqrt{(1600+900)\mathrm{cm^2}}\) \(W = 9\times 10^9\mathrm{Nm^2/c^2}\cdot 5\times 10^{-6}C\cdot 100\times 10^{-6}C \left( \frac{1}{50cm} - \frac{1}{40cm} \right)\) \(W = (9\times 10^9 \cdot 5\times 10^{-6} \cdot 100\times 10^{-6}) \left( \frac{1}{0.5m} - \frac{1}{0.4m} \right)\) \(W = \frac{9}{20}J\) So, the work done by the external force on the charge is \(\frac{9}{20}J\) which is equivalent to (C) \((81 / 20) \mathrm{J}\).

Key concepts

Electrostatics

Electrostatics is the study of electric charges at rest. It focuses on the forces, fields, and potentials associated with these charges. When dealing with point charges, we consider static electric charges which exert forces on each other. These forces can result in the creation of electric fields and potentials.
The electric potential energy arises due to the position of a charge relative to other charges. It's crucial to understand that the distribution of multiple charges can create complex electric field patterns. These fields influence the electric potential energy in a system, which can be calculated based on the location and magnitude of the charges involved.
For our exercise, the charges are stationary and positioned at certain points, allowing us to analyze their interaction purely based on their potential energy.

Coulomb's Law

Coulomb's Law is fundamental to understanding electrostatic interactions between point charges. It states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The mathematical expression is given by:

• \(F = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{r^2}\)

Here, \(q_1\) and \(q_2\) are the magnitudes of the charges, and \(r\) is the distance between them. The constant \(\frac{1}{4\pi \epsilon_0}\) represents the permittivity of free space, approximately equal to \(9 \times 10^9\,\mathrm{Nm^2/c^2}\).
This law not only helps in calculating the force but also is crucial for determining the electric potential energy, as seen in our problem. We utilized it to calculate how energy changes when charges are moved closer or further apart, affecting their potential energy.

Work and Energy

In physics, work involves transferring energy through force applied over a distance. For charges, work is done when they are moved in an electric field. Electric potential energy changes account for this work. Work done by or against electric fields shifts energy between potential and kinetic forms.
In the problem presented, this electric work is calculated by noting the changes in potential energy as the charge was moved from point B to point C. It's important to remember that work done is independent of path, due to the conservative nature of electric fields. The work is simply the difference of initial and final potential energies. This concept is encapsulated by the formula:

• \(W = U_{\text{final}} - U_{\text{initial}}\)

This formula shows how energy stored in the electric field is adjusted as the configuration of charges changes.

Vector Mathematics

When dealing with electrostatics, vector mathematics becomes a crucial tool, especially when calculating distances and forces in systems involving multiple charges. Vectors provide both magnitude and direction, which makes them ideal for representing electric fields and forces.
Our problem requires using the Pythagorean theorem, a natural application of vector mathematics, to determine the distance between charges after one is moved. This is critical because the distance between charges directly affects the system's potential energy. The theorem states:

• \(AC = \sqrt{(AB)^2 + (BC)^2} \)

Here, this calculation helps in accurately finding the new separation distance needed to ascertain the final potential energy.
Vector math enables us to rigorously calculate distances and angles in space, which then informs the potential energy calculations necessary for understanding work done during charge movement.