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When protons are accelerated through a potential difference, they gain kinetic energy, which leads to an increase in their speed. Imagine you have a proton starting from rest. As it is accelerated by a potential difference, its potential energy is converted into kinetic energy. This process can be described by the equation:

• Potential energy: \(qV\)
• Kinetic energy: \(\frac{1}{2}mv^2\)

The charge of a proton is represented by \(e\), and its mass is a known constant. By equating these two forms of energy (i.e., \(eV = \frac{1}{2}mv^2\)), we can solve for the velocity \(v\) of the protons. This is the speed they will have after being accelerated by the potential difference. Once we calculate this velocity using the given potential difference and known constants, we gain insight into how fast the proton beam is moving as it enters the magnetic field. This step is crucial since the speed of the protons will greatly influence subsequent calculations related to magnetic field interactions.

Centripetal force is the force required to keep an object moving in a circular path. For charged particles like protons moving through a magnetic field, this force is provided by the magnetic force.
As the protons enter the magnetic field, they experience a force that is perpendicular to both their motion and the magnetic field. This magnetic force keeps them moving along a circular path, ensuring they don't drift away in a straight line.
To understand this interaction, recall the equation for magnetic force:

• Magnetic force: \(qvB\)

Meanwhile, the formula for centripetal force is:

• Centripetal force: \(\frac{mv^2}{r}\)

Setting these two forces equal allows us to find the magnetic field \(B\) needed to bend the proton's path into a circle of the desired diameter, where \(r\) is the radius of the path (half the diameter). This ensures that the proton beam follows a circular trajectory inside the magnetic field.

Electrons, much like protons, also follow a circular path when subjected to a magnetic field, but there are key differences due to their significantly smaller mass. To make electrons travel in a circular path of the same size as protons, we use the same speed calculated earlier for the protons.
The expression for magnetic force and centripetal force for electrons remains similar:

• Magnetic force (electron): \(e v B_e\)
• Centripetal force: \(\frac{m_e v^2}{r}\)

Here, \(m_e\) represents the mass of an electron, which is much smaller than that of a proton. To maintain the same radius of curvature, the magnetic field \(B_e\) for electrons will differ, targeting an appropriately different magnitude due to the electron's smaller mass. By calculating \(B_e\), we adjust for these differences, enabling the electrons to navigate the same path as the protons in the magnetic field.

Potential difference, often termed as voltage, is a measure of electric potential energy per unit charge. It's like a push that accelerates charged particles, such as protons or electrons. When these particles move through a potential difference, they gain kinetic energy which manifests as an increase in their velocity.
The formula connecting potential difference \(V\) with energy is:

• Energy gained: \(qV\)

Here, \(q\) represents the charge of the particle. In this context, when a proton or an electron moves through a potential difference, its speed changes based on this energy gain.
This is the driving force behind the initial speed of particles entering a magnetic field. Understanding potential difference helps clarify how energy is transformed as charged particles accelerate, paving their way into more complex interactions, such as those involving magnetic fields.