Question

A student wants to make a very small particle accelerator using a 7.5 V battery. What speed will a proton have after being accelerated from rest through the 7.5 V potential difference? What speed will an electron have after being accelerated from rest through the 9 V potential difference?

Answer 1

The speed of a proton accelerated through a 7.5 V potential difference, and the speed of an electron accelerated through a 9 V potential difference, can be determined using the principle that the electrical potential energy is converted into kinetic energy, with the formula v = √(2eV/m).

Step-by-step explanation:

To determine the speed of a proton after being accelerated through a 7.5 V potential difference, the energy gained by the proton can be equated to its kinetic energy after acceleration. The potential energy due to the voltage V for a charge q is qV, and this is equal to the kinetic energy (1/2)mv² for non-relativistic speeds. Using the charge of a proton (e = 1.60 × 10-19 C) and its mass (1.67 × 10-27 kg), you can find the velocity using the equation v = √(2eV/m).

Applying the same principle to an electron being accelerated through a 9 V potential difference, and using the mass of an electron (9.11 × 10-31 kg), we can calculate its velocity after acceleration.

However, as the protons and electrons approach a significant fraction of the speed of light, their kinetic energy will start increasing faster than their velocity due to relativistic effects, which this simple calculation does not take into account. In such cases, a more complex relativistic formula is required to accurately calculate their speeds.