Problem 27 - University Physics with Modern Physics

Question

Before the advent of solid-state electronics, vacuum tubes were widely used in radios and other devices. A simple type of vacuum tube known as a diode consists essentially of two electrodes within a highly evacuated enclosure. One electrode, the cathode, is maintained at a high temperature and emits electrons from its surface. A potential difference of a few hundred volts is maintained between the cathode and the other electrode, known as the anode, with the anode at the higher potential. Suppose that in a particular vacuum tube the potential of the anode is 295 \(\mathrm{V}\) higher than that of the cathode. An electron leaves the surface of the cathode with zero initial speed. Find its speed when it strikes the anode.

Step-by-Step Solution

Answer

The electron's speed when it strikes the anode is approximately 3.216 million m/s.

Step 1: Understanding the Problem

We are given a vacuum tube with a potential difference of 295 V between the anode and the cathode. An electron is emitted from the cathode with zero initial speed and we need to find the speed of the electron when it reaches the anode.

Step 2: Using Energy Conservation

In this situation, we can apply the principle of energy conservation. The electron gains kinetic energy equal to the loss of electric potential energy as it moves from the cathode to the anode. The equation is \[ \Delta KE + \Delta PE = 0 \]which simplifies to \[ \frac{1}{2}mv^2 = e\Delta V \] where \( e \) is the elementary charge (\(1.6 \times 10^{-19} \text{C}\)), \( \Delta V = 295 \text{V} \), \( m \) is the electron mass (\(9.11 \times 10^{-31} \text{kg}\)), and \( v \) is the speed to find.

Step 3: Solving for Final Speed

Rearrange the equation from step 2 to solve for \( v \): \[ v = \sqrt{\frac{2e\Delta V}{m}} \] Substitute the known values: \[ v = \sqrt{\frac{2 \times 1.6 \times 10^{-19} \times 295}{9.11 \times 10^{-31}}} \] Calculate to find \( v \).

Step 4: Calculating the Result

Substituting the values into the equation gives: \[ v = \sqrt{\frac{2 \times 1.6 \times 10^{-19} \times 295}{9.11 \times 10^{-31}}} \approx \sqrt{1.033 \times 10^{13}} \approx 3.216 \times 10^6 \text{ m/s} \] Thus, the speed of the electron when it strikes the anode is approximately \(3.216 \times 10^6 \text{ m/s}.\)

Key Concepts

Electron Motion

In a vacuum tube diode, electron motion is a crucial element. Inside the tube, electrons move from the cathode towards the anode. This happens because of the potential difference maintained between the two electrodes. The cathode emits electrons when heated, a process known as thermionic emission. The emitted electrons start with virtually no speed.

As electrons travel towards the anode, they accelerate due to the force exerted by the electric field within the tube. This field is created by the potential difference, or voltage, between the cathode and anode. An electron, being negatively charged, moves towards the positively charged anode. This motion of electrons is a fundamental action that allows vacuum tubes to control electrical currents.

• Electrons are emitted from a heated cathode.
• They accelerate towards the anode due to the electric field.
• The motion is directed by the potential difference in the tube.

Energy Conservation

The principle of energy conservation plays a pivotal role in understanding how vacuum tubes work. As electrons travel from the cathode to the anode, they convert potential energy into kinetic energy. This energy transformation is a classic example of energy conservation laws at work.

In a vacuum tube diode, each electron starts with zero kinetic energy when emitted from the cathode. As it moves towards the anode, the potential energy of the electron decreases, while its kinetic energy increases. This is due to the work done on the electron by the electric field.

• The total energy of the electron remains constant.
• Potential energy lost is converted to kinetic energy.
• Understanding this conversion helps calculate the electron's speed.

Potential Difference

Potential difference, often referred to as voltage, is the driving force behind electron motion in a vacuum tube diode. This difference is maintained between the cathode and anode, with the anode being at a higher potential.

The potential difference creates an electric field inside the tube, which propels electrons towards the anode. The magnitude of this potential difference impacts the speed at which electrons move. In the given problem, a potential difference of 295 V is applied, facilitating the acceleration of the electron as it makes its journey between the electrodes.

• Voltage is the key motivator of electron movement.
• The anode is at a higher potential compared to the cathode.
• A higher potential difference results in a greater electron velocity.