What are X-rays? Electrons are produced and used in an X-ray tube - Leaving Cert Physics - Question 10 - 2017

The energy of an X-ray photon can be calculated using the formula:

$E = hf$

where:

• $h$ is Planck's constant, approximately $6.626 imes 10^{-34} ext{ Js}$,
• $f$ is the frequency calculated using the speed of light $c$ and the wavelength $ext{λ}$:

f = rac{c}{ ext{λ}}

Using the wavelength $ext{λ} = 0.02 ext{ nm} = 0.02 imes 10^{-9} ext{ m}$:

f = rac{3.00 imes 10^8 ext{ m/s}}{0.02 imes 10^{-9} ext{ m}} = 1.5 imes 10^{16} ext{ Hz}

Now substituting for $E$:

$E = (6.626 imes 10^{-34} ext{ Js})(1.5 imes 10^{16} ext{ Hz})$

Calculating gives: $E ext{ } ext{≈ } 9.9 imes 10^{-15} ext{ J}.$

The maximum velocity can be calculated using the kinetic energy formula:

$E = \frac{1}{2} mv^2$

Rearranging gives:

$v = \sqrt{\frac{2E}{m}}$

Where:

• $E$ is the energy calculated previously ($9.9 imes 10^{-15} ext{ J}$)
• $m$ is the mass of an electron, approximately $9.11 imes 10^{-31} ext{ kg}$.

Thus,

$v = \sqrt{\frac{2(9.9 imes 10^{-15})}{9.11 imes 10^{-31}}} ext{ m/s}$

Calculating yields: $v ext{ } ext{≈ } 1.48 imes 10^8 ext{ m/s}.$

The voltage applied to the electrons can be calculated from the relationship between energy and charge:

$E = qV$

Rearranging gives:

$V = \frac{E}{q}$

For an electron, the charge $q ext{ } ext{≈ } 1.6 imes 10^{-19} ext{ C}$.

Substituting:

$V = \frac{9.9 imes 10^{-15}}{1.6 imes 10^{-19}}$

Calculating yields: $V ext{ } ext{≈ } 62000 ext{ V}.$

The labelled diagram of a photocell includes the following components:

• Anode
• Semi-cylindrical cathode
• Case
• Vacuum Any additional components can be labelled as necessary.

A photocell conducts current by allowing light of suitable frequency to strike the cathode, which emits electrons. The emitted electrons move toward the anode, generating a flow of current as they gain energy from the light.

Given a current of $2 ext{ µA} = 2 imes 10^{-6} ext{ A}$, using the equation:

$Q = It$

where:

• $Q$ is charge,
• $I$ is current,
• $t$ is time in seconds (1 minute = 60 seconds):

$Q = (2 imes 10^{-6})(60) = 1.2 imes 10^{-4} ext{ C}$

Now, to find the number of electrons:

$n = \frac{Q}{e}$

Where:

• $e = 1.6 imes 10^{-19} ext{ C}$ is the charge of one electron.

Thus,

$n = \frac{1.2 imes 10^{-4}}{1.6 imes 10^{-19}} = 7.5 imes 10^{14} ext{ electrons}.$