Problem 16 - College Physics

Question

\(\bullet\) 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 in a direction perpendicular to its original direction (Fig. \(20.60 ) .\) The beam travels a distance of 1.18 cm while in the field. What is the magnitude of the magnetic field?

Step-by-Step Solution

Answer

The magnitude of the magnetic field is approximately 0.0226 T.

Step 1: Understanding the Problem

We need to find the magnitude of the magnetic field that causes the proton beam to change direction from its original direction to a perpendicular one. To achieve this, we will use the fact that protons in a magnetic field follow a circular path, with the beam's path length corresponding to a quarter of the circle since it exits perpendicular to the original direction.

Step 2: Identify Known Variables

The speed of the protons, \( v = 1.20 \) km/s, must be converted into meters per second. Therefore, \( v = 1200 \) m/s. The path or arc length \( L = 1.18 \) cm is converted into meters, \( L = 0.0118 \) m. The charge of a proton is \( q = 1.6 \times 10^{-19} \) C, and the mass of a proton is \( m = 1.67 \times 10^{-27} \) kg.

Step 3: Connecting Concepts with Circular Motion

Since the beam’s path is one quarter of a circular orbit, the relationship between the angle subtended by the path (90 degrees, or \( \frac{\pi}{2} \) radians) and the radius of the circle \( r \) is given by \( L = r \cdot \frac{\pi}{2} \). Therefore, \( r = \frac{2 \cdot L}{\pi} \).

Step 4: Apply Centripetal Motion Formula

The centripetal force needed to keep the protons in circular motion is given by \( F = \frac{mv^2}{r} \). Since this force is provided by the magnetic force \( F = qvB \) (where \( B \) is the magnetic field strength), we equate: \[ \frac{mv^2}{r} = qvB \]. Solving for \( B \), we get \( B = \frac{mv}{qr} \).

Step 5: Calculate the Radius

Substitute \( L = 0.0118 \) m into the equation for radius: \[ r = \frac{2 \cdot 0.0118}{\pi} \approx 0.00751 \text{ m}\].

Step 6: Solve for Magnetic Field Magnitude

Using the formula for \( B \), substitute known values: \[ B = \frac{(1.67 \times 10^{-27} \text{ kg})(1200 \text{ m/s})}{(1.6 \times 10^{-19} \text{ C})(0.00751 \text{ m})} \approx 0.0226 \text{ T}\].

Step 7: Final Result

After calculating the aforementioned expression, the magnitude of the magnetic field is approximately 0.0226 Tesla.

Key Concepts

Protons

Protons are subatomic particles with a positive electric charge. They are located in the nucleus of an atom, along with neutrons. Protons carry a charge of about \( +1.6 \times 10^{-19} \ \text{C} \) and possess a mass of approximately \( 1.67 \times 10^{-27} \ \text{kg} \). These small particles play a crucial role in forming matter as the number of protons in the nucleus determines the atomic number, classifying the element.
In experiments, managing beams of protons allows scientists to explore the behavior of elementary particles under various forces, such as magnetic fields. This quality makes them ideal to study electromagnetic properties because their motion is easily influenced by their positive charge. Protons have applications in medical imaging, particle accelerators, and in probing the fundamental properties of matter.

Circular Motion

When objects, such as a beam of protons, move in circular paths, they exhibit circular motion. This motion occurs when there is a constant force acting perpendicular to the velocity of the protons, deflecting them into a curved path. In the case of this exercise, we see how magnetic force causes the protons to travel in a circular arc.
A key feature of circular motion is that even if the speed is constant, the object is continuously accelerating towards the center of the circle due to the change in direction. This inward acceleration is what causes the circular motion.
Understanding this type of motion is essential in many fields, such as engineering and physics. It helps us in designing safe structures like amusement rides or understanding planetary motions around stars.

Centripetal Force

Centripetal force is the "center-seeking" force required to keep an object moving along a circular path. In the case of protons entering a magnetic field, the magnetic force acts as the centripetal force. This force doesn't exist on its own but emerges from interactions—such as friction, tension, or magnetism—that happen to point towards the circle's center.
Mathematically, centripetal force \( F_c \) is given by the formula:

• \( F_c = \frac{mv^2}{r} \)

Where:

• \( m \) is the mass of the object
• \( v \) is the velocity
• \( r \) is the radius of the path

By understanding centripetal force, students can grasp how objects like planets stay in orbit and why roller coasters are designed the way they are.

Magnetic Force

Magnetic force is a vital concept in the realm of electromagnetism. When charged particles, such as protons, move through a magnetic field, they experience a force perpendicular to both the field and their direction of motion. This movement in the presence of a magnetic field causes them to follow a circular path.
The magnitude of magnetic force \( F_m \) influencing a charged particle can be calculated using the equation:

• \( F_m = qvB \)

Which involves:

• \( q \) - the electric charge of the particle
• \( v \) - the velocity of the particle
• \( B \) - the strength of the magnetic field

Magnetic force is essential in devices like motors, generators, and magnetic storage media. By examining how proton beams are deflected in magnetic fields, scientists can deduce important properties of the field and the particle's characteristics.