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Escape speed is the minimum velocity required for an object to travel infinitely far away from a gravitational or electrostatic source without any further propulsion. This means the object can effectively "escape" the influence of the gravitational or electrostatic field it is in.
For example, in this problem, we want to find out the speed needed by an electron to escape a charged sphere's electric field. The electron, starting at rest, must reach a point far away from the sphere where its kinetic energy in the system is zero. The sphere here has a specific charge distribution, and given its radius and charge, we can determine the escape velocity.

• The notion of escape speed is closely linked to potential energy. Potential energy must be overcome for an object to escape a field.
• Escape speed is unique because it depends on the characteristics of the object exerting the field (like the sphere's charge here) and not the escaping object alone (the electron).

This calculation does not consider external factors like friction or air resistance, as it assumes a vacuum environment up to a certain distance from the field.

Electrostatic potential energy refers to the energy stored due to the positions of charged particles in an electric field. It plays a crucial role in calculating escape speed, as this energy must be overcome for the object to escape.
In our exercise, we are dealing with a charged sphere, and the electrostatic potential energy ( \( U \) ) for an electron at the surface is given by:\[U = \frac{k_e Q e}{R}\]
where:

• \( k_e \) is Coulomb's constant, \( 8.99 \times 10^9 \text{ Nm}^2 \text{C}^{-2} \).
• \( Q \) is the total charge of the sphere, \( 1.6 \times 10^{-15} \text{ C} \).
• \( e \) is the charge of an electron, \( 1.6 \times 10^{-19} \text{ C} \).
• \( R \) is the radius of the sphere, \( 0.01 \text{ m} \).

This formula calculates the potential energy due to the presence of another charge within the electric field of the sphere. This energy is what the electron needs to overcome to reach infinity (essentially escaping the sphere's electric influence for good). Understanding this concept helps us comprehend how charged particles behave in electric fields.

The law of conservation of energy is fundamental in physics and states that energy cannot be created or destroyed; it can only change forms. For the electron escape problem, this principle implies that the total mechanical energy of the system remains constant.
In the context of the escape speed, the conservation of energy principle translates to ensuring that the initial energy (combination of kinetic and potential energy at the surface of the sphere) equals the final energy (kinetic energy at infinity, is zero). The equation used here is:

• \( \frac{1}{2}mv^2 = \frac{k_e Q e}{R} \)

where:

• \( m \) is the mass of the electron, \( 9.11 \times 10^{-31} \text{ kg} \).
• \( v \) is the escape velocity we seek.

This relationship ensures that all potential energy the electron had initially is converted into kinetic energy needed to reach infinity. Solving this equation gives us the precise escape velocity needed. This concept underscores the balance and transformation of energy in any given system, providing us with a powerful tool to analyze physical phenomena.