Problem 24 - University Physics with Modern Physics

Question

A beam of protons traveling at 1.20 \(\mathrm{km} / \mathrm{s}\) enters a uniform magnetic field, traveling perpendicular to the field. The beam exits the magnetic field, leaving the field in a direction perpendicular- to its original direction (Fig. 27.48\()\) . The beam travels a distance of 1.18 \(\mathrm{cm}\) while in the field. What is the magnitude of the magnetic field?

Step-by-Step Solution

Answer

The magnetic field magnitude is approximately 2.65 T.

Step 1: Understand the Problem

A beam of protons enters a magnetic field perpendicularly, travels through it, and exits at a right angle to its original direction. This implies that the protons move in a circular path within the magnetic field. The distance traveled by the protons in the magnetic field represents a quarter of the circle's circumference.

Step 2: Use Centripetal Force Equation

Since the protons move in a circular path, the magnetic force provides the centripetal force. The equation for centripetal force is given by \( F = \frac{mv^2}{r} \), where \( m \) is the mass, \( v \) is the velocity, and \( r \) is the radius of the circle. Meanwhile, the magnetic force is given by \( F_B = qvB \), where \( q \) is charge and \( B \) is the magnetic field.

Step 3: Relate the Forces

Set the centripetal force equal to the magnetic force: \( \frac{mv^2}{r} = qvB \). This simplifies to \( r = \frac{mv}{qB} \) for the radius \( r \).

Step 4: Calculate Radius from Geometry

Since the protons travel a quarter of a circle, the distance they travel \( d \) is \( \frac{2\pi r}{4} = \frac{\pi r}{2} \). Solving for \( r \), we have \( r = \frac{2d}{\pi} \). Substitute \( d = 1.18 \times 10^{-2} \text{ m} \).

Step 5: Solve for Magnetic Field

Substituting \( r = \frac{2 \times 1.18 \times 10^{-2}}{\pi} \) into \( r = \frac{mv}{qB} \), and solving for \( B \), we get \( B = \frac{mv}{qr} \). Use the proton's mass \( m = 1.67 \times 10^{-27} \text{ kg} \), charge \( q = 1.60 \times 10^{-19} \text{ C} \), and velocity \( v = 1.20 \times 10^3 \text{ m/s} \).

Step 6: Perform the Calculation

With all values substituted: \[ r = \frac{2 \times 1.18 \times 10^{-2}}{\pi} \approx 7.51 \times 10^{-3} \text{ m} \]\[ B = \frac{1.67 \times 10^{-27} \times 1.20 \times 10^3}{1.60 \times 10^{-19} \times 7.51 \times 10^{-3}} \approx 2.65 \text{ T} \]

Step 7: Finalize the Answer

The magnitude of the magnetic field is approximately 2.65 Tesla.

Key Concepts

Proton Beam

In physics, a proton beam is essentially a group of protons, which are subatomic particles with a positive charge, moving together through space. In this problem, we are dealing with protons traveling in a beam. Protons have a specific mass and charge:

• Mass of a proton: approximately \(1.67 \times 10^{-27} \text{ kg}\)
• Charge of a proton: \(1.60 \times 10^{-19} \text{ C}\)

When a proton beam enters a magnetic field, the path it takes is crucial, especially compared to fields perpendicular to their motion. This interaction allows fascinating phenomena such as circular motion, as seen in this exercise. Understanding how protons behave in such environments is key to many applications in physics and engineering, including particle accelerators and magnetic confinement in fusion reactors. Moreover, a proton beam's speed and direction are significant parameters that influence how it interacts with external forces, like a magnetic field, to create specific motion paths.

Centripetal Force

Centripetal force is the force that keeps an object moving in a circular path. It acts towards the center of the circle. In our case of protons moving through a magnetic field, the centripetal force is critical because it determines the circular motion path of these charged particles. Now, the formula for centripetal force is essential: \[ F = \frac{mv^2}{r} \]where:

• \( m \) is the mass of the proton,
• \( v \) is the velocity, and
• \( r \) is the radius of the circle.

This formula tells us that any net force that causes circular motion is directed towards the center and is proportional to the mass of the object and the square of its velocity, inversely proportional to the radius of the circular path. In the presence of a magnetic field, this centripetal force is provided by the magnetic force, which we'll dive into next.

Magnetic Force

The magnetic force is an essential concept when dealing with charged particles like protons moving through a magnetic field. This force is responsible for altering the direction of the particle's motion without changing its speed. The magnetic force acting on a charged particle moving through a magnetic field is given by:\[ F_B = qvB \]where:

• \( q \) is the charge of the particle,
• \( v \) is its velocity, and
• \( B \) is the magnetic field strength.

The direction of the magnetic force is perpendicular to both the magnetic field and the velocity of the particle, adhering to the right-hand rule. This perpendicular direction is what causes the proton beam to move in a circular motion when entering the field at a right angle. Balancing this magnetic force with centripetal force leads to interesting scenarios where we can calculate properties like the radius of curvature and the magnetic field intensity from the physical parameters of the proton.

Circular Motion

Circular motion describes the movement of an object along the circumference of a circle. In this exercise, the proton beam undergoes circular motion due to the perpendicular entry into a magnetic field, which acts as the centripetal force. Understanding the path of circular motion requires knowing the distance traveled within the circular path. When the protons enter the magnetic field, they trace out a quarter of a circle. Therefore:\[ d = \frac{\pi r}{2} \]where:

• \( d \) is the distance traveled (1.18 cm converted to meters), and
• \( r \) is the circle's radius, which can be deduced from the geometry.

By solving this equation for \( r \), we derive the radius of the circular path, a crucial step in calculating the magnitude of the magnetic field. The fascinating part about circular motion in a magnetic field is the ability to predict the behavior of charged particles such as protons, which is essential in many scientific and industrial applications, including cyclotrons used in medicine and research.