Question

A helium leak detector uses a mass spectrometer to detect tiny leaks in a vacuum chamber. The chamber is evacuated with a vacuum pump and then sprayed with helium gas on the outside. If there is any leak, the helium molecules pass through the leak and into the chamber, whose volume is sampled by the leak detector. In the spectrometer, helium ions are accelerated and released into a tube, where their motion is perpendicular to an applied magnetic field, \(\vec{B},\) and they follow a circular path of radius \(r\) and then hit a detector. Estimate the velocity required if the radius of the ions' circular path is to be no more than \(5.00 \mathrm{~cm},\) the magnetic field is \(0.150 \mathrm{~T}\), and the mass of a helium- 4 atom is about \(6.64 \cdot 10^{-27} \mathrm{~kg}\). Assume that each ion is singly ionized (has one electron less than the neutral atom). By what factor does the required velocity change if helium-3 atoms, which have about \(\frac{3}{4}\) as much mass as helium- 4 atoms, are used?

Short answer

Answer: The required velocity for a helium-4 ion to have a circular path with a radius no more than 5 cm in a magnetic field of 0.150 T is approximately 7.23 × 10^4 m/s. If helium-3 atoms are used instead, the required velocity changes by a factor of 1.33.

Step-by-step solution

List the given information

Radius of circular path, r = 5 cm (0.05 m) Magnetic field, B = 0.150 T Mass of helium-4 atom, m_He4 = 6.64 × 10^-27 kg Charge of helium-4 ion, q = e = 1.6 × 10^-19 C (singly ionized)

Write the formula for the force on a charged particle moving in a magnetic field and the centripetal force

The formula for the force on a charged particle moving in a magnetic field is F = qvB The formula for the centripetal force is F = (m*v^2)/r

Equate the two forces and solve for the velocity of helium-4 ions

Since the force due to the magnetic field is equal to the centripetal force, we can write qvB = (m*v^2)/r We want to find the velocity, v, so we can rearrange the equation to solve for v. v = (qmB)/(m*r) Now, plug in the given values for q, m_He4, B, and r, and solve for v. v_He4 = ((1.6 × 10^-19 C) * (0.150 T))/(6.64 × 10^-27 kg * 0.05 m) v_He4 ≈ 7.23 × 10^4 m/s

Calculate the velocity of helium-3 ions and find the factor by which the velocities change

The mass of helium-3 is 3/4 the mass of helium-4. Let m_He3 be the mass of helium-3. m_He3 = (3/4) * m_He4 = (3/4) * (6.64 × 10^-27 kg) ≈ 4.98 × 10^-27 kg Now we can use the formula we derived earlier to find the velocity of helium-3 ions. v_He3 = ((1.6 × 10^-19 C) * (0.150 T))/(4.98 × 10^-27 kg * 0.05 m) v_He3 ≈ 9.65 × 10^4 m/s Finally, find the factor by which the velocities change. Factor = v_He3 / v_He4 = (9.65 × 10^4 m/s) / (7.23 × 10^4 m/s) ≈ 1.33 The required velocity changes by a factor of 1.33 if helium-3 atoms are used instead of helium-4 atoms.

Key concepts

Magnetic Field Force on Ions

Understanding how a magnetic field affects ions is crucial when examining tools like mass spectrometers in helium leak detection systems. When charged particles, like the helium ions produced in the spectrometer, enter a magnetic field at a velocity perpendicular to the field, they experience a force. This force is not random but has both magnitude and direction. It is directed perpendicular to both the velocity of the particle and the magnetic field, causing the ions to move in a circular path rather than a straight line.

This is described by the equation for the magnetic force: \( F = qvB \), where \( F \) is the force in newtons (N), \( q \) is the charge of the ion in coulombs (C), \( v \) is the velocity in meters per second (m/s), and \( B \) is the magnetic field strength in teslas (T). It's this force that bends the path of helium ions, helping to separate and detect them based on their mass-to-charge ratio. The careful calibration of the mass spectrometer allows for the precise identification of leaks, as different ions will have different circular paths based on their velocity.

Centripetal Force

Centripetal force is a concept that often goes hand-in-hand with understanding circular motion, especially in systems like mass spectrometers used for helium leak detection. It is the required force that keeps an object moving in a circular path and is always directed towards the center of the path. Centripetal force does not exist on its own but is a result of other forces, in this case, the magnetic field force.

The centripetal force can be calculated using the formula: \( F = \frac{m \times v^2}{r} \), where \( m \) represents the mass of the ion, \( v \) is the velocity, and \( r \) is the radius of the circular path. By setting the magnetic force equal to the centripetal force (since they are the same force in this context), one can derive the velocity of the charged particles. With this understanding, students can appreciate how centripetal force allows the mass spectrometer to constrain the ions to a circular trajectory, leading to their successful detection after leaks.

Velocity of Charged Particles

The velocity of charged particles plays a pivotal role in how they move within a mass spectrometer and is essential for identifying and quantifying leaks in systems such as vacuum chambers. The velocity determines how far ions curve within a magnetic field and, therefore, how they will be detected. For example, in a helium leak detector, the velocity at which helium ions must move to follow a circular path with a given radius in a known magnetic field can be calculated by rearranging the equation that equates the force on a charged particle moving in a magnetic field with centripetal force.

The resulting equation, \( v = \frac{q \times B}{m \times r} \), enables us to determine the velocity \( v \) given the charge of the ion \( q \), the magnetic field strength \( B \), the mass of the ion \( m \), and the circular path radius \( r \). It's interesting to note that the velocity will change if the mass of the particle changes, which is exactly what happens when comparing helium-4 ions to helium-3 ions. The less massive helium-3 ions require a larger velocity to maintain the same circular path, which demonstrates an inverse relationship between the mass of the ion and its velocity in magnetic fields used in helium leak detection.