Problem 13 - College Physics

Question

\(\bullet\) In a cloud chamber experiment, a proton enters a uniform 0.250 T magnetic field directed perpendicular to its motion. You measure the proton's path on a photograph and find that it follows a circular arc of radius 6.13 \(\mathrm{cm} .\) How fast was the proton moving?

Step-by-Step Solution

Answer

The proton was moving at approximately \( 1.47 \times 10^7 \ \text{m/s} \).

Step 1: Understanding the Problem

We are given a proton moving in a magnetic field. We need to find its speed based on the radius of the circular path it follows in the magnetic field.

Step 2: Identify Relevant Equation

The force acting on the proton due to the magnetic field is the centripetal force that causes it to move in a circular path. The equation that relates these is the centripetal force formula: \[ F_c = \frac{mv^2}{r} \]where \( m \) is the mass of the proton, \( v \) is the velocity, and \( r \) is the radius of the circular path. Moreover, the force due to the magnetic field on a charged particle is given by:\[ F_m = qvB \]where \( q \) is the charge, \( v \) is the velocity, and \( B \) is the magnetic field strength.

Step 3: Set Equations Equal

Since the magnetic force provides the centripetal force, we set \( F_c = F_m \). Thus:\[ \frac{mv^2}{r} = qvB \]This equation can be simplified to find the velocity \( v \):\[ v = \frac{qBr}{m} \]

Step 4: Substitute Known Values

Now, substitute the known values into the equation:- \( q \) is the charge of a proton, \( 1.6 \times 10^{-19} \) C- \( B \) is the magnetic field, 0.250 T- \( r \) is the radius, 6.13 cm = 0.0613 m- \( m \) is the mass of a proton, \( 1.67 \times 10^{-27} \) kg\[ v = \frac{(1.6 \times 10^{-19})(0.250)(0.0613)}{1.67 \times 10^{-27}} \]

Step 5: Calculate the Velocity

Calculate the velocity from the substituted values:\[ v = \frac{(1.6 \times 10^{-19})(0.250)(0.0613)}{1.67 \times 10^{-27}} \approx 1.47 \times 10^7 \ \text{m/s} \]

Key Concepts

Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. In the cloud chamber experiment, a proton moves through a magnetic field. This field is uniform and perpendicular to the proton's initial direction of motion. This is a common setup in physics experiments to study the trajectory of charged particles.
The strength of the magnetic field is measured in teslas (T). A strong magnetic field exerts a greater force on charged particles than a weak one. For example, in the given problem, a uniform magnetic field of 0.250 T is used, which influences the movement of the proton and causes it to take a circular path. This characteristic behavior is because magnetic fields exert forces that are perpendicular to the velocity of the charged particle, affecting its path significantly.
The effect of the magnetic field is captured by the formula:

• Force due to the magnetic field: \( F_m = qvB \).

Here, \( q \) represents the charge, \( v \) is velocity, and \( B \) is the magnetic field strength. This formula is crucial to linking magnetic fields with particle motion.

Centripetal Force

Centripetal force is the force required to make a body follow a curved path. Its direction is always towards the center of the circle along which the body moves. In the cloud chamber experiment, centripetal force is needed to keep the proton moving in a circular path when it travels through the magnetic field.
The proton experiences a centripetal force because the magnetic force, caused by the magnetic field, keeps pulling it towards the center of its circular path. The following equation gives the centripetal force:

• \( F_c = \frac{mv^2}{r} \)

In this equation, \( m \) is the mass of the proton, \( v \) its velocity, and \( r \) the radius of the circular path. The centripetal force equation is critical for understanding motion in magnetic fields because it underpins why the proton's path is circular.

Real-life Implications

Centripetal force isn't just a physics concept; it has real-world applications. For instance, it helps explain how cars can turn at speed without flipping, and it's the force that acts on passengers in a loop-the-loop amusement park ride. In magnetic fields and particle physics, understanding centripetal force allows us to predict and interpret particle trajectories.

Velocity of a Proton

Velocity is a vector quantity which means it has both magnitude and direction. For a proton in a magnetic field, its velocity helps determine the trajectory it will follow.
In the exercise, the velocity of the proton is determined using the formula for magnetic force and centripetal force being set equal to each other:

• \( v = \frac{qBr}{m} \)

Here, \( q \) is the charge of the proton, \( B \) is the magnetic field strength, \( r \) is the radius of the circular path, and \( m \) is the mass of the proton.
Applying these values can tell us how fast the proton is moving by calculating the magnetic and centripetal force interaction. Understanding how velocity is influenced by magnetic fields is vital for predicting the paths charged particles will take in these fields.

Proton Properties

Protons are subatomic particles with a positive charge. Their behavior in magnetic fields helps physicists understand complex atomic structures and reactions. In practical terms, determining the velocity of a proton also aids in the design of various technologies such as particle accelerators.

Circular Motion

Circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. In the context of a magnetic field, a charged particle like a proton will follow a circular path due to the magnetic forces acting upon it.
When a proton enters a perpendicular magnetic field, the magnetic force acts as a centripetal force and continuously changes the direction of the proton, resulting in circular motion. The radius of the circle can be calculated from the velocity and other known variables like charge, mass, and magnetic field strength.
The formula \( \frac{mv^2}{r} = qvB \) helps establish that this circular path is possible because the magnetic force provides a centripetal pull on the proton. By balancing these forces, the concept of circular motion becomes clear.

• The proton follows a perfect circle in a uniform magnetic field.
• The radius of the circle provides insight into the particle's energy.

Applications in Physics

Circular motion principles are foundational to devices known as cyclotrons, which are used to accelerate particles to high speeds for collision and reaction experiments. They allow researchers to explore fundamental properties of matter and interactions between particles under controlled conditions.