Question

Two point charges repel each other with a force of \(100 \mathrm{~N}\). One of the charges is increased by \(10 \%\) and other is reduced by \(10 \%\). The new force of repulsion at the same distance would be \(\ldots \ldots \mathrm{N}\). (A) 121 (B) 100 (C) 99 (D) 89

Short answer

The new force of repulsion at the same distance after altering the charges is \(99 N\), which corresponds to the answer (C).

Step-by-step solution

Write down the formula for Coulomb's law

The formula for the force between two point charged objects at distance r can be given by Coulomb's Law: \[F = k \frac{q_1 q_2}{r^2}\] Where \(F\) is the force, \(k\) is the Coulomb's constant (\(8.99 \times 10^9 N m^2 C^{-2}\)), \(q_1\) and \(q_2\) are the charges, and \(r\) is the distance between them.

Find the relation between initial charges and force

Let the initial charges be \(q_1\) and \(q_2\), and the force between them is 100 N. According to Coulomb's law, the initial force can be written as: \[F_1 = k \frac{q_1 q_2}{r^2} = 100 N\]

Modify the charges and write the new force equation

Now, we increase one charge by 10% and decrease the other charge by 10%. The new charges will be: \[q_1' = 1.1q_1\] \[q_2' = 0.9q_2\] Now, we can write the equation for the new force between the charges at the same distance \(r\): \[F_2 = k \frac{q_1' q_2'}{r^2}\]

Calculate the new force

Replacing the new charges in the above equation, we get: \[F_2 = k \frac{1.1q_1 * 0.9q_2}{r^2}\] \[F_2 = k \frac{0.99q_1 q_2}{r^2}\] We know that the initial force equation is \(F_1 = k \frac{q_1 q_2}{r^2} = 100 N\). Thus, we can write the new force equation in terms of the initial force equation: \[F_2 = 0.99F_1\] \[F_2 = 0.99(100 N)\]

Find the new force value

Calculating the new force, we get the final value: \[F_2 = 99 N\] So, the new force of repulsion at the same distance after altering charges is 99 N, which corresponds to the answer (C).

Key concepts

Point Charges

Point charges are fundamental concepts in electrostatics and are treated as infinitesimally small charged particles that possess a certain quantity of electric charge. In reality, point charges help us simplify and understand electric forces in idealized system. These are used to model the electric field and force interactions between particles when their physical sizes are negligible compared to the distances between them.
Some essential characteristics of point charges include:

• They are considered to have all their charge concentrated at a single point in space.
• Their interactions can be mathematically described using Coulomb's Law.
• They are effectively dimensionless within physics problems, allowing for easier calculations.

Understanding point charges is an essential step to progressing in the study of electrostatics and allows us to dive deeper into concepts like electric fields and force vectors.

Electric Force

Electric force is an essential force in physics that describes the interaction between charged objects. This force is exponentially significant since it governs how objects with electric charges move and interact. According to Coulomb's Law, electric forces can either be attractive or repulsive.
Some key points about electric force:

• It exists between any two objects with charge and can be calculated using Coulomb's Law: \[ F = k \frac{q_1 q_2}{r^2} \]Where:
  • \( F \) is the magnitude of the force
  • \( k \) is Coulomb's constant
  • \( q_1 \) and \( q_2 \) are the magnitudes of the charges
  • \( r \) is the distance between them
• It acts along the line joining the charges, which can be a direct line of attraction or repulsion.
• The direction of the force is attractive if the charges are opposite and repulsive if they are the same.

The concept of electric force is integral to comprehending how charged particles influence one another, leading to various phenomena in electricity and magnetism.

Charge Manipulation

Charge manipulation involves changing the magnitude or distribution of electric charge on a particle or a system. In the earlier example, the exercise demonstrates how charge manipulation affects the electric force between two point charges, and it shows how slight changes in charge impact the resultant forces due to Coulomb's Law.
Here are a few important points about charge manipulation:

• Changing the charge on a particle alters the electric force it experiences; for instance, increasing a charge leads to proportional increases in the resulting force.
• In our exercise, the increase and decrease of each charge by 10% demonstrate how forces relationally multiply or reduce when charge magnitude changes.
• This process frequently involves understanding ratios; in this problem, the net factor affecting the force came down to \(0.99\) because it was product of the changes, \(1.1 \times 0.9 = 0.99\).

Charge manipulation shows its importance in various scientific and engineering applications, such as fine-tuning the design of electronic devices and managing static electricity in industrial processes.