Understanding the Concepts
* Potential Difference (Voltage): A potential difference (voltage) is the difference in electric potential energy per unit charge between two points. When a charged particle moves through a potential difference, it gains or loses kinetic energy.
* Work-Energy Theorem: The work done on an object equals the change in its kinetic energy. In this case, the work done on the proton by the electric field is equal to the potential energy it loses.
* Charge of a Proton: The charge of a proton is +1.602 x 10⁻¹⁹ Coulombs (C).
Calculation
1. Potential Energy Loss:
The potential energy lost by the proton is equal to the product of its charge and the potential difference:
ΔPE = q * ΔV = (1.602 x 10⁻¹⁹ C) * (25000 V) = 4 x 10⁻¹⁵ J
2. Kinetic Energy Gain:
According to the work-energy theorem, the potential energy lost is equal to the kinetic energy gained:
ΔKE = ΔPE = 4 x 10⁻¹⁵ J
3. Final Kinetic Energy:
Since the proton starts from rest, its initial kinetic energy is zero. Therefore, its final kinetic energy is:
KE_final = 4 x 10⁻¹⁵ J
4. Final Speed:
The kinetic energy is related to the mass (m) and speed (v) of the proton by:
KE_final = (1/2) * m * v²
Solving for the final speed, we get:
v = √(2 * KE_final / m)
The mass of a proton is 1.672 x 10⁻²⁷ kg. Plugging in the values:
v = √(2 * 4 x 10⁻¹⁵ J / 1.672 x 10⁻²⁷ kg) ≈ 2.19 x 10⁷ m/s
Result
The proton gains a speed of approximately 2.19 x 10⁷ m/s after falling through a potential difference of 25,000 volts.
