Two electrons each moving with a velocity of 10₆ m s⁻¹ are released towards each other. What will be the closest distance of approach between them?

• A: 1.53 × 10⁻⁸ m
• B: 2.53 × 10⁻¹⁰ m
• C: 2.53 × 10⁻⁶ m
• D: zero

To determine the closest distance of approach between two electrons moving towards each other, we can use the concept of electric potential energy and kinetic energy. Each electron has a charge of approximately -1.6 × 10⁻¹⁹ C and a mass of about 9.11 × 10⁻³¹ kg.

Step-by-Step Calculation

Kinetic Energy of Each Electron

The kinetic energy (KE) of each electron can be calculated using the formula:

KE = 0.5 * m * v²

Substituting the values:

• m = 9.11 × 10⁻³¹ kg
• v = 10⁶ m/s

Thus, the kinetic energy for one electron is:

KE = 0.5 * (9.11 × 10⁻³¹) * (10⁶)² = 4.55 × 10⁻¹³ J

Total Kinetic Energy

Since there are two electrons, the total kinetic energy is:

Total KE = 2 * 4.55 × 10⁻¹³ J = 9.1 × 10⁻¹³ J

Electric Potential Energy at Closest Approach

The electric potential energy (PE) between two charges is given by:

PE = k * |q₁ * q₂| / r

Where:

• k = 8.99 × 10⁹ N m²/C² (Coulomb's constant)
• q₁ = q₂ = -1.6 × 10⁻¹⁹ C
• r = distance of closest approach

Setting Kinetic Energy Equal to Potential Energy

At the closest approach, all kinetic energy is converted to potential energy:

9.1 × 10⁻¹³ J = (8.99 × 10⁹) * (1.6 × 10⁻¹⁹)² / r

Solving for r

Rearranging the equation gives:

r = (8.99 × 10⁹ * (1.6 × 10⁻¹⁹)²) / (9.1 × 10⁻¹³)

Calculating this yields:

r ≈ 2.53 × 10⁻¹⁰ m

Final Answer

The closest distance of approach between the two electrons is:

B: 2.53 × 10⁻¹⁰ m