How strong an electric field is needed to accelerate electrons in an X-ray tube from rest to one-tenth the speed of light in a distance of \(5.0 \mathrm{cm} ?\)

When we talk about accelerating electrons, we're looking at how charged particles gain speed when they interact with an electric field. Just like when you slide down a playground slide, where gravity is pulling you down, an electric field can ‘pull’ on an electron, which has a negative charge.

An important aspect of this concept is understanding that the acceleration doesn't happen endlessly; there’s a specific distance or region where this acceleration takes place—in our case, the 5.0 cm in the X-ray tube. Since electrons have a known charge, we can calculate the influence of an electric field over this distance. By applying a precise force over a certain distance, we can precisely control the speed of the electrons.

In our specific example, we want the electron to reach one-tenth the speed of light. To do this, we first need to define the final velocity we're aiming for, which based on the speed of light (about 300,000 km/s), means the electrons must reach 30,000 km/s. We then use physical formulas to calculate the strength of the electric field necessary to achieve this velocity over the given distance.

At speeds approaching the speed of light, classical physics takes a back seat to Einstein’s theory of relativity. This means that the energy of an object, particularly a small particle like an electron, doesn’t just depend on its mass and velocity (as Newtonian kinematics would suggest), but also on how fast it's going relative to the speed of light. This is what we call ‘relativistic kinetic energy’.

In the regular kinetic energy formula, you might multiply half the mass of an object by the square of its velocity. But when the object is an electron speeding up to a significant fraction of the speed of light, the calculation is not that straightforward. The mass of the electron seems to increase due to relativity. Therefore, we have to use the energy-momentum relation, incorporating the rest mass of the electron and taking into account the ‘relativistic mass’ increase, to figure out the total energy, and consequently, the relativistic kinetic energy.

It’s the change in this kinetic energy that we want to equal the work done by the electric field, allowing us to figure out exactly how strong the electric field needs to be for our scenario.

The physics of an X-ray tube revolves around transferring energy to electrons so that they can cause photons to be emitted, which we use as X-rays. The core principle involves a high-voltage electric field to accelerate electrons which then collide with a metal target. When these high-speed electrons decelerate upon collision, they emit energy in the form of X-rays.

This process can be thought of like throwing a ball into the air. As it climbs, it slows down right before it starts falling back to Earth. That slow-down transfers energy to the air (though it's not exactly noticeable). In an X-ray tube, the ‘slow-down’ of electrons is much more dramatic, and that's where the energy comes out as X-ray photons, which are then used in medical imaging and other applications.

So in order to generate these X-rays, the electrons need to be moving quickly enough when they hit the target. Calculating the electric field strength, as outlined in the original exercise, ensures the electrons have the necessary energy for X-ray production when they reach the end of their acceleration path.