Two electrons (each with mass m = 9.1 × 10−31 kg and
charge q = −1.6 × 10−19 C) are located along the circum-
ference of an imaginary circle with radius R = 95 cm. One
is fixed in place at the top of the circle (i.e., the 12 o’clock
position), while the other is placed on the right side of the
circle (i.e., the 3 o’clock position) and then released from
rest. What is the maximum speed v that the second elec-
tron will reach when it is infinitely far away from the first
electron? The symbolic answer should be stated in terms
of m, q, R, θ, and k, though not all of these quantities may
be needed.

This problem involves calculating the maximum speed \( v \) that the second electron will reach when it is infinitely far away from the first electron, using principles from electrostatics and energy conservation.

### Step 1: Energy conservation
We can apply the conservation of mechanical energy. Initially, when the second electron is released, it has only electrostatic potential energy due to its interaction with the first electron, and no kinetic energy (since it's released from rest). As the second electron moves infinitely far away, the electrostatic potential energy becomes zero, and all the initial potential energy is converted into kinetic energy.

The total mechanical energy \( E_{\text{total}} \) of the system is conserved:

\[
E_{\text{initial}} = E_{\text{final}}
\]

Where:
- \( E_{\text{initial}} \) is the initial electrostatic potential energy,
- \( E_{\text{final}} \) is the final kinetic energy.

### Step 2: Initial electrostatic potential energy
The electrostatic potential energy between two charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by:

\[
U = k \frac{q_1 q_2}{r}
\]

In this case:
- \( q_1 = q_2 = q = -1.6 \times 10^{-19} \, \text{C} \),
- \( r = R = 0.95 \, \text{m} \) (the distance between the two electrons),
- \( k = 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \) (Coulomb's constant).

Thus, the initial potential energy is:

\[
U_{\text{initial}} = k \frac{q^2}{R}
\]

### Step 3: Final kinetic energy
When the second electron is infinitely far away from the first, the potential energy between them is zero, and the total energy is purely kinetic. The kinetic energy of the second electron is:

\[
K_{\text{final}} = \frac{1}{2} m v^2
\]

### Step 4: Equating energies
Since mechanical energy is conserved:

\[
U_{\text{initial}} = K_{\text{final}}
\]

Substituting the expressions for \( U_{\text{initial}} \) and \( K_{\text{final}} \):

\[
k \frac{q^2}{R} = \frac{1}{2} m v^2
\]

### Step 5: Solving for the speed \( v \)
Rearrange the equation to solve for \( v \):

\[
v = \sqrt{\frac{2k q^2}{m R}}
\]

### Final Answer:
The maximum speed \( v \) that the second electron will reach is:

\[
v = \sqrt{\frac{2k q^2}{m R}}
\]

### Variables:
- \( m \) is the mass of the electron,
- \( q \) is the charge of the electron,
- \( R \) is the radius of the circle,
- \( k \) is Coulomb’s constant.

---

Would you like more details on any part of this solution, or have any specific questions?

### Follow-up Questions:
1. How would the result change if the electron was placed at a different position on the circle before release?
2. What would happen if the charge of one of the electrons were different?
3. How does the radius \( R \) affect the final speed?
4. Could this method be applied to particles other than electrons? How?
5. What would the solution look like in the presence of a uniform external electric field?

### Tip:
Always check the consistency of units when applying physical formulas to ensure they match the expected result.

Two electrons (each with mass m = 9.1 × 10−31 kg and charge q = −1.6 × 10−19 C) are located along the circumference of an imaginary circle with radius R = 95 cm. One is fixed in place at the top of the circle (i.e., the 12 o’clock position), while the other is placed on the right side of the circle (i.e., the 3 o’clock position) and then released from rest. What is the maximum speed v that the second electron will reach when it is infinitely far away from the first electron? The symbolic answer should be stated in terms of m, q, R, θ, and k, though not all of these quantities may be needed.

Discover how to calculate the maximum speed of an electron released from rest in an electrostatic system. Using the principles of energy conservation and Coulomb's Law, this physics problem involves the mass, charge, and distance between two electrons along a circular path. Suitable for undergraduate physics students.

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