91Ó°ÊÓ

Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique. Concept Questions Particle 1 and particle 2 carry the same charge \(q,\) but particle has a smaller mass than particle \(2 .\) These two particles accelerate from rest through the same electric potential difference \(V\) and enter the same magnetic field, which has a magnitude \(B\). The particles travel perpendicular to the field on circular paths. Upon entering the field region, which particle, if either, has the greater (a) kinetic energy and (b) speed? Give your reasoning. Problem The masses of the particles are \(m_{1}=2.3 \times 10^{-8} \mathrm{~kg}\) and \(m_{2}=5.9 \times 10^{-8} \mathrm{~kg} .\) The radius of the circular path for particle 1 is \(r_{1}=12 \mathrm{~cm} .\) What is the radius of the circular path for particle \(2 ?\)

Step by step solution

01

Analyze Kinetic Energy Concept

When a charged particle is accelerated from rest through an electric potential difference \( V \), its gain in kinetic energy \( K \) is given by \( K = qV \). Since both particles have the same charge \( q \) and are subjected to the same potential difference \( V \), they both gain the same kinetic energy. Thus, both particles 1 and 2 have equal kinetic energy upon entering the magnetic field.

02

Analyze Speed Concept

The speed \( v \) of a particle after accelerating through a potential difference \( V \) is determined by \( \frac{1}{2}mv^2 = qV \). For particles with identical charge \( q \), the particle with less mass \( m \) achieves a higher speed. Therefore, particle 1, which has a smaller mass, will have a higher speed than particle 2.

03

Relate Radius to Mass and Speed

In a magnetic field, the radius \( r \) of a particle's path is given by \( r = \frac{mv}{qB} \). Solving for \( v \) using the energy equation \( v = \sqrt{\frac{2qV}{m}} \), we can substitute \( v \) into this equation. Therefore, the radius \( r \) becomes \( r = \frac{m}{qB} \sqrt{\frac{2qV}{m}} = \sqrt{\frac{2m^2V}{q^2B^2}} \). Notice that \( r \) depends on \( \sqrt{m} \), meaning particle 2 with higher mass should have a larger path radius.

04

Calculate Radius for Particle 2

Using the formula for radius, for particle 1 we have \( r_1 = \sqrt{\frac{2m_1^2V}{q^2B^2}} \). Substituting the known values into \( r_1 = 12 \) cm, get the expression \( 12 = \sqrt{\frac{2(2.3 \times 10^{-8})^2V}{q^2B^2}} \). For particle 2, \( r_2 = \sqrt{\frac{2m_2^2V}{q^2B^2}} = \sqrt{\frac{2(5.9 \times 10^{-8})^2V}{q^2B^2}} \). Since \( r_2/r_1 = m_2/m_1 \), \( r_2 = 12 \times (5.9/2.3) \approx 30.8 \) cm.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Energy

When a charged particle accelerates from rest due to an electric potential difference, it gains kinetic energy. This energy, denoted by \( K \), can be described by the equation:

• \( K = qV \)

where \( q \) is the charge of the particle and \( V \) is the electric potential difference.

This formula demonstrates that the kinetic energy gained by a charged particle is solely dependent on its charge and the electric potential through which it travels, and not on its mass or speed. Accordingly, when two particles with identical charges are accelerated through the same potential difference, they will both acquire the same kinetic energy.

In this exercise, since both particles 1 and 2 have the same charge and potential difference, they reach the same region of the magnetic field with equal kinetic energy.

Magnetic Field

A magnetic field exerts a force on moving charged particles, affecting their trajectory. For a charged particle moving perpendicularly to a magnetic field, the force \( F \) experienced can be expressed as:

• \( F = qvB \)

where \( q \) is the charge, \( v \) is the velocity, and \( B \) is the magnetic field strength.

This force causes the charged particle to follow a circular path within the magnetic field. Since both particles maneuver perpendicular to the magnetic field in this scenario, they experience a centripetal force that affects their motion.

Magnetic fields are crucial in applications like cyclotrons and mass spectrometers, where particles' paths need to be controlled and predicted based on their charge and velocity.

Circular Motion

When a charged particle enters a magnetic field at a perpendicular angle, it undergoes circular motion. The particle's path is governed by the balance of two forces: the magnetic force providing the centripetal force needed for circular motion. The radius \( r \) of this path is given by:

• \( r = \frac{mv}{qB} \)

where \( m \) is the particle's mass, \( v \) is its speed, \( q \) is its charge, and \( B \) is the magnetic field strength.

Given that both particles started from rest, the speed after accelerating in an electric potential difference \( V \) can be derived using \( v = \sqrt{\frac{2qV}{m}} \). This speed dependency implies that the radius of the circular path primarily depends on the mass of the particle, with heavier particles tracing larger radii due to their lower velocities for the same kinetic energy.

Understanding this relationship helps explain why particle 2, being heavier, follows a larger radius path than particle 1.

Mass-to-Charge Ratio

The mass-to-charge ratio \( \frac{m}{q} \) of a particle is a crucial parameter in determining its behavior in electric and magnetic fields. Specifically, in a magnetic field, the path radius of a particle for a given charge and velocity is reliant on this ratio:

• \( r \propto \sqrt{\frac{m}{q}} \)

This relationship indicates that a greater mass-to-charge ratio results in a larger radius of the circular path.

For practical applications like isotope separation or in analyzing atomic particles, the mass-to-charge ratio can be determined by measuring the radius of curvature of particle trajectories.

In the given exercise, particle 2 had a larger mass and thus a larger mass-to-charge ratio compared to particle 1, confirming its greater radius during circular motion in the magnetic field.