Problem 19 - College Physics

Question

\(\bullet\) A beam of protons is accelerated through a potential dif- ference of 0.745 \(\mathrm{kV}\) and then enters a uniform magnetic field traveling perpendicular to the field. (a) What magnitude of field is needed to bend these protons in a circular arc of diameter 1.75 \(\mathrm{m} ?\) (b) What magnetic field would be needed to produce a path with the same diameter if the particles were electrons having the same speed as the protons?

Step-by-Step Solution

Answer

(a) 0.37 T for protons. (b) 0.00028 T for electrons.

Step 1: Determine the Kinetic Energy of the Protons

When protons are accelerated through a potential difference, they gain kinetic energy equal to the electrical energy: \[ KE = qV \]where \( q \) is the charge of the proton \( (1.6 \times 10^{-19} \text{ C}) \), and \( V \) is the potential difference \( (0.745 \times 10^3 \text{ V}) \). Calculate the kinetic energy: \[ KE = 1.6 \times 10^{-19} \text{ C} \times 0.745 \times 10^3 \text{ V} = 1.192 \times 10^{-16} \text{ J} \]

Step 2: Calculate the Speed of the Protons

The kinetic energy is also given by the expression: \[ KE = \frac{1}{2}mv^2 \]where \( m \) is the mass of a proton \( (1.67 \times 10^{-27} \text{ kg}) \) and \( v \) is the speed. Set the two expressions for kinetic energy equal and solve for \( v \): \[ \frac{1}{2}mv^2 = 1.192 \times 10^{-16} \text{ J} \]\[ v = \sqrt{\frac{2 \times 1.192 \times 10^{-16}}{1.67 \times 10^{-27}}} \approx 3.08 \times 10^6 \text{ m/s} \]

Step 3: Calculate the Required Magnetic Field for Protons

The magnetic force provides the centripetal force necessary to keep the protons in a circular path: \[ qvB = \frac{mv^2}{r} \]where \( r \) is the radius of the path \( (0.875 \text{ m}) \) since the diameter is 1.75 m, and \( B \) is the magnetic field. Solve for \( B \): \[ B = \frac{mv}{qr} = \frac{1.67 \times 10^{-27} \times 3.08 \times 10^6}{1.6 \times 10^{-19} \times 0.875} \approx 0.37 \text{ T} \]

Step 4: Calculate Speed of Electrons with Given Protons' Speed

The problem states that the electrons have the same speed as the protons. Since we've calculated the speed for protons as \( 3.08 \times 10^6 \text{ m/s} \), electrons will also have this speed.

Step 5: Calculate the Required Magnetic Field for Electrons

Using the same circular motion equation for electrons, substitute the mass of the electron \( (9.11 \times 10^{-31} \text{ kg}) \) while the speed and radius are unchanged:\[ qvB = \frac{mv^2}{r} \]Solve for \( B \) with electron parameters:\[ B = \frac{mv}{qr} = \frac{9.11 \times 10^{-31} \times 3.08 \times 10^6}{1.6 \times 10^{-19} \times 0.875} \approx 2.80 \times 10^{-4} \text{ T} \]

Key Concepts

Kinetic Energy

Kinetic energy is a fundamental concept in physics. It refers to the energy an object possesses due to its motion. When accelerating charged particles, such as protons and electrons, the energy acquired through motion is termed kinetic energy.
The formula used for kinetic energy is:

• \( KE = \frac{1}{2}mv^2 \)

Here, \( m \) represents mass, and \( v \) stands for velocity of the particle.
In our problem, protons acquire kinetic energy as they move through an electric field. This energy translates from the potential difference provided, calculated with:

• \( KE = qV \)

where \( q \) is the charge, and \( V \) is the potential difference. With a known potential change, this energy can be computed for each particle. This relationship ensures that energy from voltage encourages acceleration of the charged particles.

Circular Motion

Circular motion describes a particle moving in a path around a central point. For charged particles like protons and electrons entering a magnetic field, their paths can bend into circular arcs.
The key to understanding this is recognizing the role of magnetic fields in affecting a charged particle's trajectory. When they enter perpendicularly, a magnetic force acts perpendicular to the velocity, encouraging a circular route.
The radius of this circular path is linked with the particle's properties and motion characteristics. In our case study, we calculate the path's diameter to understand the necessary conditions for its trajectory. This predictive modeling assists in designing how magnetic fields can bend paths predictably.

Centripetal Force

Centripetal force is crucial in maintaining circular motion for particles within a magnetic field. This inward-directed force ensures particles remain on their curved paths without veering off.
For particles like protons and electrons, the centripetal force is provided by the magnetic field acting on charged particles. This relationship can be expressed as:

• \( qvB = \frac{mv^2}{r} \)

This equation reveals the balance between magnetic force and centripetal necessity. The force driving circular motion must equal the mass times velocity squared over the radius.
This equation assists in deducing the magnetic field's strength necessary to produce a particular circular path by manipulating variables such as particle mass, velocity, and radius.

Potential Difference

Potential difference, or voltage, is essentially the electric pressure that drives charge flow through a circuit. In our context, it's the electric power source fuelling proton movement through space.
When protons speed up through a specific potential difference, they gain kinetic energy directly proportional to that voltage change. It's pivotal in converting electric potential energy into the kinetic form necessary for motion.
The relationship \( KE = qV \) highlights how any change in potential difference impacts kinetic energy acquired. Hence, calculating the influence of potential difference is key to understanding the resultant behavior of charged particles, whether they be protons or electrons.