Two identical 20g particles each carry 4.0μC of charge. One is held fixed, and the other is placed 1.0 mm away and released.

Find the speed of the moving charge when it's 1.0 cm from the fixed charge.

Find the speed of the moving charge when it's 1.0 km from the fixed charge.

To find the speed of the moving charge at a distance of 1.0 cm from the fixed charge, we can use the concept of electrostatic potential energy.

Step 1: Calculate the potential energy of the system at the initial position.
The potential energy (U) of two charges can be calculated using the formula:
U = k * (q1 * q2) / r,
where k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2), q1 and q2 are the charges in Coulombs, and r is the distance between the charges in meters.

U = (8.99 x 10^9 Nm^2/C^2) * [(4.0 x 10^-6 C) * (4.0 x 10^-6 C)] / (1.0 x 10^-3 m)
U = 143600 J

Step 2: Calculate the potential energy of the system at the final position.
The potential energy can be calculated in the same way, using the new distance of 1.0 cm (0.01 m).

U_final = (8.99 x 10^9 Nm^2/C^2) * [(4.0 x 10^-6 C) * (4.0 x 10^-6 C)] / (0.01 m)
U_final = 1.436 x 10^7 J

Step 3: Calculate the kinetic energy of the moving charge.
The difference in potential energy can be considered as the change in potential energy (ΔU) and is equal to the kinetic energy (K) of the moving charge.

ΔU = U_final - U
ΔU = (1.436 x 10^7 J) - (143600 J)
ΔU = 1.42 x 10^7 J

K = ΔU

Step 4: Calculate the speed of the moving charge.
The kinetic energy (K) can be calculated using the formula:
K = (1/2) * m * v^2,
where m is the mass of the charged particle and v is the speed of the particle.

Since the particles are identical, we can consider the mass (m) of the particle as 20 g (0.02 kg).

K = (1/2) * (0.02 kg) * v^2
1.42 x 10^7 J = 0.01 kg * v^2
v^2 = (1.42 x 10^7 J) / (0.01 kg)
v^2 = 1.42 x 10^9 m^2/s^2
v ≈ sqrt(1.42 x 10^9) ≈ 37,697 m/s

Therefore, the speed of the moving charge when it is 1.0 cm from the fixed charge is approximately 37,697 m/s.

To find the speed of the moving charge when it is 1.0 km from the fixed charge, we can use the same steps:

Step 1: Calculate the potential energy of the system at the initial position (1.0 mm).
U = (8.99 x 10^9 Nm^2/C^2) * [(4.0 x 10^-6 C) * (4.0 x 10^-6 C)] / (1.0 x 10^-3 m)
U = 143600 J

Step 2: Calculate the potential energy of the system at the final position (1.0 km = 1000 m).
U_final = (8.99 x 10^9 Nm^2/C^2) * [(4.0 x 10^-6 C) * (4.0 x 10^-6 C)] / (1000 m)
U_final = 143.6 J

Step 3: Calculate the kinetic energy of the moving charge.
ΔU = U_final - U
ΔU = 143.6 J - 143600 J
ΔU = -143456 J

K = ΔU

Step 4: Calculate the speed of the moving charge.
K = (1/2) * (0.02 kg) * v^2
-143456 J = 0.01 kg * v^2
v^2 = (-143456 J) / (0.01 kg)
v^2 = -1.43456 x 10^7 m^2/s^2 (negative value indicates a deceleration)
v ≈ sqrt(-1.43456 x 10^7) ≈ -3788 m/s

Therefore, the speed of the moving charge when it is 1.0 km from the fixed charge is approximately -3788 m/s.

To find the speed of the moving charge at different distances from the fixed charge, we can use the principle of conservation of mechanical energy. This principle states that the sum of potential energy and kinetic energy remains constant in the absence of non-conservative forces, such as friction.

Let's begin by finding the initial distance and potential energy of the system when the moving charge is released from 1.0 mm away.

1. First, convert the distance from millimeters to meters:
1.0 mm = 0.001 m

2. The initial distance (r1) between the charges is 0.001 m.
The initial potential energy (U1) is given by the formula:
U1 = k * (q1 * q2) / r1

where k is the electrostatic constant, q1 and q2 are the charges, and r1 is the initial distance.

Substituting the values into the equation:
U1 = (9 × 10^9 N m^2/C^2) * ((4.0 × 10^(-6) C)^2) / 0.001 m

Calculating U1:
U1 = 7.2 J

Note: The potential energy is always positive, indicating the repulsive nature of the charges.

Next, let's find the final distance and potential energy of the system when the moving charge is 1.0 cm away.

1. Convert the distance from centimeters to meters:
1.0 cm = 0.01 m

2. The final distance (r2) between the charges is 0.01 m.
The final potential energy (U2) is given by the formula:
U2 = k * (q1 * q2) / r2

Substituting the values into the equation:
U2 = (9 × 10^9 N m^2/C^2) * ((4.0 × 10^(-6) C)^2) / 0.01 m

Calculating U2:
U2 = 0.72 J

Since the total mechanical energy is conserved, the sum of the initial potential energy and kinetic energy equals the sum of the final potential energy and kinetic energy:

U1 + K1 = U2 + K2

For both cases, the initial kinetic energy (K1) is zero because the moving charge is released from rest. Therefore, the equation becomes:

U1 = U2 + K2

We can rearrange the equation to find the final kinetic energy (K2):

K2 = U1 - U2

Now we can calculate the final kinetic energies for both cases.

For the first case when the moving charge is 1.0 cm away:

K2 = 7.2 J - 0.72 J
= 6.48 J

The final kinetic energy is 6.48 J.

To find the speed (v), we can use the formula:

K = (1/2) * m * v^2

Rearranging the formula to solve for v:

v = √(2 * K / m)

Substituting the values into the equation:

v = √(2 * 6.48 J / 0.02 kg)
= √(648 J/kg)
≈ 25.45 m/s

Therefore, the speed of the moving charge when it is 1.0 cm from the fixed charge is approximately 25.45 m/s.

For the second case when the moving charge is 1.0 km away:

K2 = 7.2 J - 0.72 J
= 6.48 J

v = √(2 * 6.48 J / 0.02 kg)
= √(648 J/kg)
≈ 25.45 m/s

Therefore, the speed of the moving charge when it is 1.0 km from the fixed charge is also approximately 25.45 m/s.