Question

Two charged spheres are \(8 \mathrm{~cm}\) apart. They are moved closer to each other enough that the force on each of them increases four times. How far apart are they now?

Short answer

Answer: The new distance between the charged spheres is 4 cm.

Step-by-step solution

Write down the initial conditions and the variables to find

The initial distance between the charged spheres is \(8 \mathrm{~cm}\). We are looking for the new distance after the force increases four times. Let's denote the initial distance as \(d_1\) and the final distance as \(d_2\). We have: - \(d_1 = 8 \mathrm{~cm}\) - Find \(d_2\)

Write down the Coulomb's Law formula

The formula for the force between two charged particles according to Coulomb's Law is: $$F = k \frac{q_1q_2}{d^2}$$ Where: - \(F\) is the force between the two charged particles. - \(k\) is the Coulomb's constant (\(8.99 \times 10^9 \mathrm{Nm^2C^{-2}}\)) - \(q_1\) and \(q_2\) are the charges of the particles. - \(d\) is the distance between the charged particles.

Use the ratio of the forces given in the problem

We are given that the force increases by four times after the spheres are moved closer. Let's set up a ratio to find the relationship between the initial and final distances: $$\frac{F_1}{F_2} = \frac{4}{1}$$ Since these forces depend on the distance between the spheres according to Coulomb's Law, we can plug in the formula for each force: $$\frac{k\frac{q_1q_2}{d_1^2}}{k\frac{q_1q_2}{d_2^2}} = \frac{4}{1}$$

Simplify the equation and solve for \(d_2\)

Notice that \(k\) and \(q_1q_2\) are present in both the numerator and the denominator, so we can cancel them out to simplify the equation: $$\frac{1}{d_1^2} \div \frac{1}{d_2^2} = 4$$ Now cross-multiply and solve for \(d_2\): $$4d_2^2 = d_1^2$$ We know \(d_1 = 8 \mathrm{~cm}\), so we can substitute that into our equation and solve for \(d_2\): $$4d_2^2 = (8 \mathrm{~cm})^2$$ $$d_2^2 = \frac{64 \mathrm{~cm}^2}{4}$$ $$d_2^2 = 16 \mathrm{~cm}^2$$ Now, take the square root of both sides to find \(d_2\): $$d_2 = \sqrt{16 \mathrm{~cm}^2}$$ $$d_2 = 4 \mathrm{~cm}$$

State the final answer

After moving the charged spheres closer to each other such that the force on each increases four times, they are now \(4 \mathrm{~cm}\) apart.

Key concepts

Electric Charge

Electric charge is a fundamental property of matter. It causes particles to experience a force when placed in an electromagnetic field. There are two types of electric charges: positive and negative.
The concept of electric charge is integral to understanding electromagnetic interactions. Electrons carry a negative charge, while protons carry a positive charge. Typically, objects have a balance of both, resulting in a neutral charge. However, when this balance is disrupted, objects become charged, leading to attractive or repulsive interactions.
Common units for measuring electric charge include coulombs, and a single electron carries approximately \(1.6 \times 10^{-19}\) coulombs. Understanding electric charge is vital for studying electrostatic phenomena such as the forces between charged objects.

• Objects with like charges repel each other.
• Conversely, objects with opposite charges attract each other.

Electrostatic Force

Electrostatic force refers to the force between charged objects. Described by Coulomb's Law, it plays a crucial role in various physical phenomena. Coulomb’s Law quantifies this force, stating that it is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance separating them.
The formula for electrostatic force is given by \(F = k \frac{q_1q_2}{d^2}\), where:

• \(F\) is the electrostatic force.
• \(k\) is Coulomb’s constant, approximately \(8.99 \times 10^9 \text{Nm}^2/ ext{C}^2\).
• \(q_1\) and \(q_2\) are the charges.
• \(d\) is the distance between the center of the two charges.

This force can either be attractive or repulsive. It depends on the nature of the charges:

• Like charges create a repulsive force.
• Unlike charges create an attractive force.

In the context of the original problem, an increase in electrostatic force by four times indicates that the distance between the charges must have changed according to Coulomb's principle, altering the interaction between them.

Distance Between Charges

The distance between charges is a critical factor in determining the strength of the electrostatic force. According to Coulomb's Law, the force is inversely proportional to the square of the distance \(d\) between the charges. This means that as the distance between the charges decreases, the force increases significantly, and vice versa.
For instance, if the distance is halved, the electrostatic force is quadrupled. This relationship is exemplified in the problem where the initial distance was reduced from 8 cm to 4 cm, resulting in a fourfold increase in force.

• The inverse square law underlines that electrostatic force changes faster than the changes in distance. It's important to calculate changes accurately when assessing forces in real-world scenarios.
• Understanding how distance affects force is crucial for solving physics problems involving charged bodies, whether it's spheres as in this example, or other charged systems.

Accurate measurement and prediction are essential in applications ranging from electrical engineering to understanding natural phenomena.