The electron beam in a color TV is accelerated through 32kV and then strikes the screen . What is the wavelength of the most energetic x_ray photon ?

To find the wavelength of the most energetic X-ray photon produced when an electron beam is accelerated through a potential difference of 32 kV, we can use the relationship between energy and wavelength. The energy gained by the electrons can be calculated using the formula for electric potential energy, and then we can relate that energy to the wavelength of the emitted X-ray photon.

Calculating the Energy of the Electron Beam

When electrons are accelerated through a potential difference (V), they gain kinetic energy equal to the work done on them by the electric field. This energy can be calculated using the formula:

E = eV

Where:

• E is the energy in joules (J).
• e is the elementary charge, approximately 1.6 x 10^-19 C.
• V is the potential difference in volts (V).

For a potential difference of 32 kV (or 32,000 V), the energy can be calculated as follows:

E = (1.6 x 10^-19 C)(32,000 V) = 5.12 x 10^-15 J

Relating Energy to Wavelength

The energy of a photon is also related to its wavelength through the equation:

E = hc/λ

Where:

• h is Planck's constant, approximately 6.626 x 10^-34 J·s.
• c is the speed of light, approximately 3.00 x 10⁸ m/s.
• λ is the wavelength in meters (m).

Finding the Wavelength

To find the wavelength, we can rearrange the equation to solve for λ:

λ = hc/E

Now, substituting the values we have:

λ = (6.626 x 10^-34 J·s)(3.00 x 10⁸ m/s) / (5.12 x 10^-15 J)

Calculating this gives:

λ ≈ 3.88 x 10^-11 m

Converting to Nanometers

Since wavelengths of X-rays are often expressed in nanometers (nm), we can convert meters to nanometers by multiplying by 10⁹:

λ ≈ 3.88 x 10^-11 m x 10⁹ nm/m = 0.0388 nm

Summary of Results

The wavelength of the most energetic X-ray photon produced by an electron beam accelerated through 32 kV is approximately 0.0388 nm. This wavelength falls within the X-ray region of the electromagnetic spectrum, indicating that the energy from the accelerated electrons is sufficient to produce high-energy photons.