Question

A particle detector utilizes a solenoid that has 550 turns of wire per centimeter. The wire carries a current of 22 A. A cylindrical detector that lies within the solenoid has an inner radius of \(0.80 \mathrm{~m} .\) Electron and positron beams are directed into the solenoid parallel to its axis. What is the minimum momentum perpendicular to the solenoid axis that a particle can have if it is to be able to enter the detector?

Short answer

Answer: The minimum perpendicular momentum required for electrons and positrons to enter the detector is approximately \(6.06 \times 10^{-23}\,\mathrm{kg\cdot m/s}\).

Step-by-step solution

Find the magnetic field inside the solenoid

Using Ampere's law, the magnetic field inside a solenoid of length \(l\) is given by the equation \(B = \mu_0 n I\), where \(B\) is the magnetic field strength, \(\mu_0\) is the permeability of free space (\(4 \pi \times 10^{-7}\, \mathrm{T \cdot m / A}\)), \(n\) is the number of turns per length, and \(I\) is the current. In this case, the solenoid has 550 turns per centimeter (or 55000 turns per meter) and carries a current of 22 A. We can use this information to find the magnetic field inside the solenoid.

Find the magnetic force acting on the particles

The magnetic force acting on a charged particle moving in a magnetic field is given by the Lorentz force equation, \(F = q(v \times B)\), where \(F\) is the magnetic force, \(q\) is the charge of the particle, \(v\) is the velocity vector of the particle, and \(B\) is the magnetic field vector. Since we are looking for the minimum perpendicular momentum for a particle to enter the detector, we want the force exerted by the magnetic field to equal the centripetal force needed for a particle to travel in a circular path with a radius equal to the inner radius of the detector. The centripetal force equation is given by \(F_c = \frac{m v^2}{r}\), where \(m\) is the mass of the particle, and \(r\) is the radius of the circle.

Finding the minimum perpendicular momentum

To find the minimum perpendicular momentum, we can equate the magnetic force and centripetal force in terms of momentum. Using the relation \(p = mv\) (momentum equals mass times velocity), we can rewrite the centripetal force equation as \(F_c = \frac{p^2}{m r}\). Similarly, the magnetic force in terms of momentum can be written as \(F = q(\frac{p}{m} \times B)\). Equating these two forces, we get the equation: $$\frac{p^2}{m r} = q\left(\frac{p}{m} \times B\right)$$ To find the minimum perpendicular momentum, we can solve this equation for \(p\): $$p = q \cdot B \cdot r$$ Now we can plug in the values for \(q\), \(B\), and \(r\). It's important to note that the charge of both an electron and a positron is \(e = 1.6\times10^{-19}\,\mathrm{C}\).

Calculate the minimum perpendicular momentum

Using the given values, we can calculate the minimum perpendicular momentum needed for a particle to enter the detector: \(p = e \cdot B \cdot r = (1.6 \times 10^{-19} \,\mathrm{C}) \cdot B \cdot (0.80 \,\mathrm{m})\) We previously found that \(B = \mu_0 n I = (4\pi \times 10^{-7} \,\mathrm{T \cdot m / A}) (55000 \,\mathrm{turns/m}) (22 \,\mathrm{A})\), so we can substitute this value of \(B\) into the equation for \(p\): \(p = (1.6 \times 10^{-19} \,\mathrm{C}) \cdot [(4\pi \times 10^{-7} \,\mathrm{T \cdot m / A}) (55000 \,\mathrm{turns/m}) (22 \,\mathrm{A})] \cdot (0.80 \,\mathrm{m})\) Calculating the value, we find the minimum perpendicular momentum that a particle can have in order to enter the detector:

= 6.06 \times 10^{-23} \,\mathrm{kg \cdot m/s}$

Key concepts

Ampere's Law

Ampere's Law is a fundamental principle used to determine the magnetic field generated by an electric current. It is particularly useful when dealing with symmetrical configurations like solenoids. The law states that the magnetic field around a closed loop is proportional to the electric current passing through the loop's area.
In mathematical terms, for a solenoid, the magnetic field inside can be calculated using the formula:

• \( B = \mu_0 n I \)

where

• \( B \) is the magnetic field strength.
• \( \mu_0 \) is the permeability of free space \( (4 \pi \times 10^{-7}\, \mathrm{T \cdot m / A}) \).
• \( n \) is the number of turns per length \( (\mathrm{turns/m}) \).
• \( I \) is the current in amperes.

Hence, in our solenoid operated detector, the identification of turns per centimeter and the amount of current allows for a straightforward calculation of the magnetic field, crucial for understanding the forces acting on the particles.

Lorentz Force

The Lorentz Force describes the force experienced by charged particles moving through a magnetic field. It's an essential concept for understanding the dynamics of particles in magnetic fields, such as in particle detectors. This force causes the charged particles to move in circular or spiral paths.
The force can be determined using the equation:

• \( F = q(v \times B) \)

Where:

• \( F \) is the magnetic force.
• \( q \) is the charge of the particle.
• \( v \) is the velocity vector of the particle.
• \( B \) is the magnetic field vector.

In the context of our exercise, electrons and positrons are subjected to this force as they enter the detector, influencing their paths. The Lorentz force needs to be balanced with the centripetal force for them to successfully curve into the detector's circular path.

Centripetal Force

Centripetal force is the force required to keep a particle moving in a circular path. In magnetic fields, particles naturally tend to follow circular trajectories due to the Lorentz force. For this continuous motion, the centripetal force must equate to the magnetic force.
The formula for centripetal force is:

• \( F_c = \frac{m v^2}{r} \)

Where:

• \( F_c \) is the centripetal force.
• \( m \) is the mass of the particle.
• \( v \) is the velocity of the particle.
• \( r \) is the radius of the circular path.

In our scenario, equating centripetal and Lorentz forces aids in establishing the relationship needed to determine the minimum perpendicular momentum of particles that can successfully enter the detector.

Solenoids

Solenoids are coil arrangements used to create uniform magnetic fields in a specific region. They consist of numerous turns of wire wound closely together, generating a magnetic field when current flows through them.
When analyzing solenoids, particularly those used in detectors, some important characteristics include:

• The uniform magnetic field inside a long solenoid is a significant advantage, providing a consistent environment for particle paths.
• Field strength is easily controlled by adjusting the current or modifying the coil's turns per unit length.
• They play a crucial role in applications like MRI machines or particle accelerators due to their ability to control magnetic fields precisely.

In the exercise's solenoid, it provides a uniform magnetic field critical for guiding particles within the cylindrical detector, demonstrating solenoids' practical utility in scientific instruments.

Momentum in Magnetic Fields

When particles move through magnetic fields, their momentum changes direction but not magnitude, leading to fascinating dynamics, particularly in circular paths.
The minimum perpendicular momentum required for a particle to move into the detector can be calculated using the insight from previous concepts by setting the centripetal force equal to the magnetic force:

• \( p = q \cdot B \cdot r \)

Where:

• \( p \) is the momentum.
• \( q \) is the charge of the particle.
• \( B \) is the magnetic field.
• \( r \) is the radius of the path.

This relationship encapsulates the effect of a magnetic field on particle momentum, demonstrating how magnetic setups can be optimized to monitor or manipulate particles effectively, with practical applications in accelerators and detectors.