Question

Два одинаковых металлических шарика заряжены положительными зарядами 5 нКл и 20 нКл. Центры шариков находятся в вакууме на расстоянии 10 см друг от друга. Шарики привели в соприкосновение. На какое расстояние после этого нужно развести их центры, чтобы сила взаимодействия осталась прежней.Помогите пожалуйста, если можно с подробным решением

Answer 1

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Answer 2

Problem Analysis

We have two identical metal spheres with positive charges of 5 nC and 20 nC. The centers of the spheres are initially 10 cm apart in a vacuum. We need to find the distance between their centers after they are brought into contact, such that the force of interaction remains the same.

Solution

To solve this problem, we can use the principle of conservation of charge. When the two spheres are brought into contact, charge is transferred between them until they reach a state of equilibrium. In this state, the total charge is conserved, but it is redistributed between the spheres.

Let's assume that after contact, the charge on the spheres is distributed in the ratio of their initial charges. The charge on the first sphere will be given by:

q1 = (q1_initial * q2_initial) / (q1_initial + q2_initial)

Similarly, the charge on the second sphere will be:

q2 = (q1_initial * q2_initial) / (q1_initial + q2_initial)

Now, let's calculate the new distance between the centers of the spheres. We can use Coulomb's law to find the force of interaction between the spheres before and after contact. Coulomb's law states that the force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

Before contact, the force of interaction between the spheres is given by:

F_initial = (k * q1_initial * q2_initial) / r_initial^2

After contact, the force of interaction between the spheres should remain the same. Therefore, the force of interaction after contact can be calculated using the new distance between the centers of the spheres:

F_final = (k * q1 * q2) / r_final^2

Since the force of interaction remains the same, we can equate the initial and final forces:

(k * q1_initial * q2_initial) / r_initial^2 = (k * q1 * q2) / r_final^2

Simplifying the equation, we get:

r_final = sqrt((q1_initial * q2_initial * r_initial^2) / (q1 * q2))

Now, let's substitute the given values into the equation and calculate the final distance between the centers of the spheres.

Calculation

Given: - q1_initial = 5 nC - q2_initial = 20 nC - r_initial = 10 cm

Using the equation:

r_final = sqrt((q1_initial * q2_initial * r_initial^2) / (q1 * q2))

Substituting the values:

r_final = sqrt((5 nC * 20 nC * (10 cm)^2) / ((q1_initial * q2_initial) / (q1_initial + q2_initial)))

Simplifying the equation:

r_final = sqrt((5 nC * 20 nC * (10 cm)^2) / ((5 nC * 20 nC) / (5 nC + 20 nC)))

r_final = sqrt((5 nC * 20 nC * (10 cm)^2) / ((5 nC * 20 nC) / (25 nC)))

r_final = sqrt((5 nC * 20 nC * (10 cm)^2) / (100 nC))

r_final = sqrt((5 * 20 * (10 cm)^2) / 100) cm

r_final = sqrt((5 * 20 * 100) / 100) cm

r_final = sqrt(100) cm

r_final = 10 cm

Answer

After the spheres are brought into contact, the distance between their centers needs to be 10 cm to maintain the same force of interaction.

Conclusion

In this problem, we calculated the distance between the centers of two identical metal spheres after they are brought into contact, such that the force of interaction remains the same. We used the principle of conservation of charge and Coulomb's law to derive the equation for the final distance. By substituting the given values into the equation, we found that the final distance between the centers of the spheres is 10 cm.