Find the relation between velocity and potential difference using classical mechanics for (a) electrons and (b) protons accelerated in vacuo. For what voltages do the velocities become onc-tenth of the velocity of light?

Understanding the kinetic energy of charged particles like electrons and protons is fundamental in physics. Kinetic energy is the energy that a particle possesses due to its motion.

When a particle accelerates, its kinetic energy increases. For charged particles in an electric field, this acceleration is the result of the electric force exerted on the charge. The formula for kinetic energy (K.E.) is \( K.E. = \frac{1}{2} m v^2 \), where \(m\) is the mass of the particle, and \(v\) is its velocity.

For instance, considering an electron with a known charge and mass, when it accelerates through a potential difference in a vacuum, its kinetic energy arises purely from the electric field's work. The greater the potential difference, the higher the velocity the electron will achieve upon exiting the electric field.

The electric potential difference, often referred to as voltage (\( V \)), is a measure of the work done by an external force in moving a charge from one point to another within an electric field without acceleration.

The electric potential difference between two points is the energy required per unit charge to move the charge between these points. It is the electric field's ability to do work on a charge and is measured in volts (\( V \)).

The relationship between voltage and the work done on a particle is given by the equation \( W = qV \), where \(W\) is the work, \(q\) is the charge of the particle, and \(V\) is the potential difference. For charged particles in motion, this work translates into kinetic energy.

The relationship between a charged particle's velocity and the electric potential difference through which it has been accelerated is a cornerstone of electromagnetism. This relationship is encapsulated by equating the particle's kinetic energy to the energy it gains from moving through a potential difference.

In the case of a particle with charge \(q\) moving through a potential difference \(V\), the kinetic energy is equivalent to the work done by the electric field: \( \frac{1}{2}mv^2 = qV \). By rearranging this equation, we can express velocity (\(v\)) as a function of the potential difference:

\( v = \sqrt{\frac{2qV}{m}} \)

This equation demonstrates that the velocity of a charged particle is proportional to the square root of the potential difference it has traversed if we consider the mass (\(m\)) and the charge (\(q\)) to be constants. Consequently, for higher potential differences, charged particles will achieve greater velocities.

Classical mechanics and electromagnetism are two important branches of physics that intersect when analyzing the motion of charged particles in an electric field. According to classical mechanics, the motion of objects can be predicted by Newton’s laws. When applied to charged particles, like electrons and protons, classical mechanics integrates with the concepts of electromagnetism, specifically the impact of electric and magnetic fields on particle dynamics.

Using classical mechanics, we derive formulas to calculate the velocity of charged particles under the influence of electromagnetic forces. This is especially relevant for particles in accelerators and cathode ray tubes, where electric fields are used to influence their paths and velocities. The simple formula for velocity derived from potential difference, as mentioned earlier, is a prime example of the application of classical mechanics principles to electromagnetic phenomena.

Moreover, understanding the classic velocity-potential relationship is crucial in a multitude of practical applications, including the design of electronic devices, medical equipment like X-ray machines, and experimental physics research, where the control over particle velocities is essential.