Two metal spheres, each of mass 1 gm and radius 0.5 cm are placed on a smooth, horizontal, insulated plate such that their centers are 15 cm apart. One is charged to a potential 500 volt and the other to 1000 volt. What is the velocity of each sphere when they drift away to a distance of 30 cm. between their centers?

Ans please

To find the velocity of each sphere when they drift away to a distance of 30 cm between their centers, we can use the principle of conservation of energy.

First, let's calculate the potential energy between the spheres at their initial position. The potential energy (PE) between two point charges can be calculated using the formula:

PE = K * (q1 * q2) / r,

where K is the electrostatic constant (K = 9 x 10^9 Nm^2/C^2), q1 and q2 are the charges of the spheres, and r is the distance between their centers.

Given:
Mass of each sphere = 1 gm = 0.001 kg
Radius of each sphere = 0.5 cm = 0.005 m
Initial distance between the centers of the spheres (r) = 15 cm = 0.15 m

The charge on the first sphere is 500 volts, and the charge on the second sphere is 1000 volts. However, voltage is potential difference, not charge. We need to convert the voltage into charge using the formula:

Charge (q) = Capacitance (C) * Voltage (V),

where Capacitance (C) can be calculated for a conducting sphere using the formula:

C = 4πε₀r,

where ε₀ is the permittivity of free space (ε₀ = 8.85 x 10^-12 F/m).

Let's calculate the charges on the two spheres:

For the first sphere:
C = 4πε₀r = 4π(8.85 x 10^-12 F/m)(0.005 m) ≈ 0.0002206 F
q1 = C * V1 = 0.0002206 F * 500 V ≈ 0.1103 C

For the second sphere:
C = 4πε₀r = 4π(8.85 x 10^-12 F/m)(0.005 m) ≈ 0.0002206 F
q2 = C * V2 = 0.0002206 F * 1000 V ≈ 0.2206 C

Now, let's calculate the initial potential energy (PE_initial) between the spheres:

PE_initial = K * (q1 * q2) / r = (9 x 10^9 Nm^2/C^2) * (0.1103 C * 0.2206 C) / 0.15 m ≈ 727.403 Nm

Next, let's calculate the final potential energy (PE_final) when the spheres are separated by a distance of 30 cm = 0.3 m:

PE_final = K * (q1 * q2) / r = (9 x 10^9 Nm^2/C^2) * (0.1103 C * 0.2206 C) / 0.3 m ≈ 484.935 Nm

Since energy is conserved, the kinetic energy (KE_final) of the system at 30 cm separation will be equal to the initial potential energy:

KE_final = PE_final = 484.935 J

The total kinetic energy (KE) of the two spheres can be calculated using the formula:

KE = (1/2) * (m1 * v1^2) + (1/2) * (m2 * v2^2),

where m1 and m2 are the masses of the two spheres, and v1 and v2 are their velocities, respectively.

Since the masses of both spheres are the same, we can rewrite the equation as:

KE = (1/2) * m * (v1^2 + v2^2)

Let's solve for the velocities, v1 and v2.

v1^2 + v2^2 = 2 * KE / m [Divide both sides by m]

v1^2 + v2^2 = 2 * 484.935 J / 0.001 kg [Substituting the values]

v1^2 + v2^2 = 969870 J/kg [Simplifying]

Now, let's evaluate the velocity of each sphere when they drift away to a distance of 30 cm between their centers. We assume this drift occurs in a straight line.

The distance traveled by each sphere is half of the total separation distance, which is 0.15 m. Let's denote this distance as d.

d = 0.15 m

The time taken to cover this distance, t, is the same for both spheres since they drift away from each other at the same time.

Now, using the formula:

d = (1/2) * a * t^2,

where a is the acceleration we need to solve for, we can express the acceleration in terms of v1 and v2:

a = (v2^2 - v1^2) / (2 * d) [Substituting v1^2 + v2^2 = 969870 J/kg]

a = (969870 J/kg - v1^2) / (2 * d)

Using Newton's second law of motion, F = m * a, and the formula for electrostatic force, F = K * (q1 * q2) / r^2, we can equate the magnitudes of these forces:

m * a = K * (q1 * q2) / r^2

Substituting the given values:

0.001 kg * a = (9 x 10^9 Nm^2/C^2) * (0.1103 C * 0.2206 C) / (0.15 m)^2

Simplifying this equation, we can solve for the acceleration, a:

a = (9 x 10^9 Nm^2/C^2) * (0.1103 C * 0.2206 C) / (0.001 kg * (0.15 m)^2)

Now that we have the value of the acceleration, we can substitute it back and solve for the velocities, v1 and v2:

v1^2 = v2^2 - 2 * a * d [Rearranging the equation for v1^2]

v2^2 = v1^2 + 2 * a * d [Rearranging the equation for v2^2]

v1 = √(v2^2 - 2 * a * d) [Taking the square root on both sides]

Now, we can substitute the values of v2, a, and d to solve for v1.

To find the velocity of each sphere when they drift away to a distance of 30 cm between their centers, we can use the principle of conservation of energy.

1. Let's start by calculating the electric potential energy of each sphere. The electric potential energy between two point charges is given by the equation:

Potential Energy = (k * q1 * q2) / r

where k is the electrostatic constant (k = 9 * 10^9 N.m^2/C^2), q1 and q2 are the charges on the spheres, and r is the distance between their centers.

2. For the first sphere with a charge of 500 volts, the potential energy is given by:

Potential Energy 1 = (k * q1 * q2) / r = (9 * 10^9) * (1 * 10^-3) * (1 * 10^-3) / 0.15

Please note that we convert the charge from volts to coulombs by multiplying by 10^-3.

3. Similarly, for the second sphere with a charge of 1000 volts, the potential energy is given by:

Potential Energy 2 = (k * q1 * q2) / r = (9 * 10^9) * (1 * 10^-3) * (1 * 10^-3) / 0.15

4. Now, let's consider the initial potential energy when the spheres are 15 cm apart, and the final potential energy when they are 30 cm apart. Since the spheres are moving against the electric potential, the work done against the electric field converts into kinetic energy.

Initial Potential Energy = Potential Energy 1 + Potential Energy 2

= (9 * 10^9) * (1 * 10^-3) * (1 * 10^-3) / 0.15 + (9 * 10^9) * (1 * 10^-3) * (1 * 10^-3) / 0.15

Final Potential Energy = (9 * 10^9) * (1 * 10^-3) * (1 * 10^-3) / 0.3 + (9 * 10^9) * (1 * 10^-3) * (1 * 10^-3) / 0.3

5. The difference between the initial and final potential energies is equal to the kinetic energy gained by the spheres during their motion. So, we can write:

Kinetic Energy = Initial Potential Energy - Final Potential Energy

6. The kinetic energy acquired by each sphere can be given to them as:

Kinetic Energy = (1/2) * m * v^2

where m is the mass of the sphere and v is its velocity.

7. Rearranging the equation, we can find the velocity of each sphere:

v = sqrt((2 * Kinetic Energy) / m)

Substituting the value of the kinetic energy we obtained in step 5 and the mass of the spheres:

v = sqrt((2 * (Initial Potential Energy - Final Potential Energy)) / 0.001)

By following these steps and plugging in the given values into the equations, you should be able to calculate the velocity of each sphere when they drift away to a distance of 30 cm.