Estimate the speed with which electrons emitted from a heated emitter of an evacuated tube impinge on the collector maintained at a potential difference of 500 V with respect to the emitter. Ignore the small initial speeds of the electrons. The specific charge of the electron, i.e., its e/m is given to be 1.76 × 1011 C kg–1. (b) Use the same form

To estimate the speed of electrons emitted from a heated emitter in an evacuated tube and impinging on a collector at a potential difference of 500 V, we can use the principles of energy conservation and the relationship between electric potential energy and kinetic energy. Let's break this down step by step.

Understanding the Energy Conversion

When electrons are accelerated through a potential difference (V), they gain kinetic energy equal to the work done on them by the electric field. The relationship can be expressed as:

• Kinetic Energy (KE) = eV

Where:

• e is the charge of the electron (approximately 1.6 × 10^-19 C).
• V is the potential difference (in this case, 500 V).

Calculating the Kinetic Energy

Substituting the values into the equation, we find the kinetic energy gained by the electron:

• KE = eV = (1.6 × 10^-19 C) × (500 V) = 8.0 × 10^-17 J.

Relating Kinetic Energy to Speed

The kinetic energy of an electron can also be expressed in terms of its mass (m) and speed (v) as follows:

• Kinetic Energy (KE) = (1/2)mv²

We can set the two expressions for kinetic energy equal to each other:

• (1/2)mv² = eV.

Finding the Mass of the Electron

To solve for the speed, we need the mass of the electron. The mass (m) of an electron is approximately 9.11 × 10^-31 kg. Now we can rearrange the equation to solve for speed (v):

• v = sqrt((2eV)/m).

Substituting Values

Now, substituting the known values into the equation:

• v = sqrt((2 × (1.6 × 10^-19 C) × (500 V)) / (9.11 × 10^-31 kg)).

Calculating this gives:

• v = sqrt((1.6 × 10^-19 C × 1000) / (9.11 × 10^-31 kg))
• v = sqrt((1.6 × 10^-16 J) / (9.11 × 10^-31 kg))
• v = sqrt(1.76 × 10¹⁴ m²/s²)
• v ≈ 1.32 × 10⁷ m/s.

Final Result

Thus, the speed of the electrons when they reach the collector at a potential difference of 500 V is approximately 1.32 × 10⁷ m/s. This high speed illustrates how effectively electric fields can accelerate charged particles, such as electrons, in a vacuum.

In summary, by applying the principles of energy conservation and using the relationship between electric potential energy and kinetic energy, we can estimate the speed of electrons in an evacuated tube. This approach not only provides a clear answer but also reinforces the fundamental concepts of electromagnetism and particle dynamics.