An electron starts from the cathode of a vacuum tube with negligible velocity. What are its kinetic energy and velocity at the anode if the potential difference between electrodes is \(100 \mathrm{~V}\) ? How many electrons per unit volume at the anode are there if \(1 \mathrm{~mA}\) is collected by every square \(\mathrm{cm}\) of electrode? Explain how it is that the current in such an electron beam is constant when vin Eq. (1.12) is increasing across the tube.

Understanding the role of potential difference in electron dynamics is crucial to comprehending the behavior of electrons in electric fields. A potential difference, often measured in volts (V), is the driving force that accelerates charged particles like electrons. When an electron moves from a negatively charged electrode (cathode) to a positively charged one (anode) through a vacuum tube, it experiences an acceleration due to the potential difference between the electrodes.

Imagine potential difference as a slope on a hill: just as a ball rolls down a hill because of the height difference (gravitational potential difference), an electron moves from the cathode to the anode due to the electric potential difference. As the electron travels, it converts potential energy into kinetic energy, and this conversion continues until the electron reaches the anode, at which point all the potential energy has been transformed.

Kinetic energy (\(E_k\text{)\text{is)} the observable result of the acceleration process due to a potential difference and can be determined with the classic physics formula \(E_k = \frac{1}{2}m_ev^2\), where \(m_e\) is the mass of an electron and \(v\) is its velocity. For a charged particle like an electron, the same amount of energy that was previously held as electrical potential energy when it was near the cathode is now present as kinetic energy when it reaches the anode.

Since we typically consider an electron starting with negligible velocity at the cathode, it's assumed that all potential energy converted is directly observable as kinetic energy, with no energy lost to other forms like heat.

To calculate an electron's velocity at the anode, we use its kinetic energy and mass. The relationship provided by kinetic energy can be rearranged to solve for velocity, employing the equation \(v = \sqrt{\frac{2E_k}{m_e}}\). By substituting the value for kinetic energy and the known mass of an electron, we arrive at the velocity. This demonstrates a key principle in physics: the amount of work or energy transfer expresses itself in the motion of the particle.

The steps for calculating electron velocity highlight the physiological transformation of potential energy (provided by the potential difference) into the kinetic domain—revealing how fast an electron is moving when it hits the anode.

Determining the concentration of electrons at the anode requires a different approach. Here we tie in the concept of electric current—an expression of the flow rate of charge. By using the current and the area of the anode, along with the charge of an electron and its velocity, we can calculate how densely packed the electrons are when they reach the anode.

The equation \(n = \frac{I}{eAv}\) represents the relationship between current (\(I\)), electron charge (\(e\)), the area across which the current is distributed (\(A\)), and the velocity of the electrons (\(v\)). Therefore, electron concentration is a direct measure of how many electrons are present in a given volume at a certain point in time—essential for understanding electron behavior in the device.

Electric current is essentially a flow of electric charge, and in our case, this charge is carried by electrons. It's measured in amperes (A), where one ampere equals the flow of one coulomb of charge per second. An important point of confusion for learners is that despite an increase in individual electron velocity, the current remains the same. This may seem counterintuitive, but it's because current depends not just on velocity but also on the number of charges passing a point per unit time.

As electrons accelerate and their velocities increase, assuming the concentration of electrons doesn't change, they will spread out along the tube, which maintains a steady flow of charge—thus a constant current. If you imagine cars moving on a highway, even as they speed up, the number of cars passing beneath a bridge every minute (analogous to current) can remain the same if they maintain consistent separation.