Problem 3 - College Physics

Question

\(\bullet\) A proton traveling at 3.60 \(\mathrm{km} / \mathrm{s}\) suddenly enters a uniform magnetic field of 0.750 \(\mathrm{T}\) , traveling at an angle of \(55.0^{\circ}\) with the field lines (Figure 20.57\()\) . (a) Find the magnitude and direction of the force this magnetic field exerts on the proton. (b) If you can vary the direction of the proton's velocity, find the magnitude of the maximum and minimum forces you could achieve, and show how the velocity should be oriented to achieve these forces. (c) What would the answers to part (a) be if the proton were replaced by an electron traveling in the same way as the proton?

Step-by-Step Solution

Answer

(a) Force on proton: \(3.70 \times 10^{-16} \mathrm{N}\), direction perpendicular to \(\mathbf{v}\) and \(\mathbf{B}\). (b) Maximum: \(4.32 \times 10^{-16} \mathrm{N}\), Minimum: \(0 \mathrm{N}\). (c) Same magnitude for electron, opposite direction.

Step 1: Understanding Lorentz Force

The force exerted by a magnetic field on a moving charge is given by the Lorentz Force Formula: \[ \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) \]where \(q\) is the charge of the particle, \(\mathbf{v}\) is its velocity, and \(\mathbf{B}\) is the magnetic field. The magnitude of this force is given by \[ F = qvB\sin\theta \]where \(\theta\) is the angle between \(\mathbf{v}\) and \(\mathbf{B}\). We will use this to find the force on the proton.

Step 2: Calculating Proton's Force Magnitude

First, calculate the magnitude of the force exerted on the proton. The charge of a proton is \(1.60 \times 10^{-19} \mathrm{C}\), velocity \(v = 3.60 \times 10^3 \mathrm{m/s}\), magnetic field \(B = 0.750 \mathrm{T}\), and \(\theta = 55.0^{\circ}\).Using \( F = qvB\sin\theta \), we have:\[ F = (1.60 \times 10^{-19} \mathrm{C})(3.60 \times 10^3 \mathrm{m/s})(0.750 \mathrm{T})\sin(55.0^{\circ}) \]\[ F \approx 3.70 \times 10^{-16} \mathrm{N} \]

Step 3: Direction of the Force

The direction of the force is perpendicular to both the velocity \(\mathbf{v}\) and the magnetic field \(\mathbf{B}\). This direction can be determined using the right-hand rule, which tells us the force will be out of the plane defined by \(\mathbf{v}\) and \(\mathbf{B}\).

Step 4: Maximum and Minimum Force

The maximum force occurs when \(\sin\theta = 1\) (i.e., \(\theta = 90^{\circ}\)) and the minimum force occurs when \(\sin\theta = 0\) (i.e., \(\theta = 0^{\circ}\) or \(180^{\circ}\)).- Maximum Force: \[ F_{max} = qvB = (1.60 \times 10^{-19} \mathrm{C})(3.60 \times 10^3 \mathrm{m/s})(0.750 \mathrm{T}) \] \[ F_{max} \approx 4.32 \times 10^{-16} \mathrm{N} \]- Minimum Force: \[ F_{min} = 0 \mathrm{N} \] To achieve these, orient the velocity perpendicular for maximum, and parallel or anti-parallel for minimum.

Step 5: Electron with Same Conditions

If the proton is replaced by an electron (with the same velocity and in the same magnetic field), the magnitude of the force calculated remains the same because the magnitude of charge \(|q|\) is the same, but the force direction reverses due to the negative charge of the electron.Therefore, \[ F = 3.70 \times 10^{-16} \mathrm{N} \] and it is opposite in direction to that experienced by the proton.

Key Concepts

Understanding Magnetic Fields

Magnetic fields are invisible forces that can exert a push or pull on certain materials, particularly affecting charged particles like protons and electrons. These fields are represented by field lines, which show the direction and strength of the magnetic force.

• The closer the lines, the stronger the magnetic field.
• Magnetic fields are measured in Tesla (\( \mathrm{T} \)), a unit that indicates the field's strength.

When a charged particle moves through a magnetic field, it experiences a force known as the Lorentz force. The magnetic field interacts with the moving charge, altering its path depending on its velocity, direction, and the angle at which it enters the field.
Understanding this interaction is critical for predicting how particles like protons will behave when subjected to magnetic fields.

Proton Motion in a Magnetic Field

The behavior of a proton in a magnetic field can be understood through its charge and motion. Protons have a positive charge and when they move through a magnetic field, they experience a force that is perpendicular to both their velocity and the magnetic field direction.
For a proton traveling at 3.60 \( \mathrm{km/s} \)n with a magnetic field strength of 0.750 \( \mathrm{T} \), and entering at an angle of \( 55.0^{\circ} \), we can determine the magnitude of the force using:\[ F = qvB\sin\theta \]Substituting the given values, we find that the proton experiences a force of about \( 3.70 \times 10^{-16} \mathrm{N} \).

• If you vary the direction of the velocity to make the angle \( \theta \) equals \( 90^{\circ} \) (perpendicular), you'll achieve the maximum force.
• Conversely, if \( \theta \) is \( 0^{\circ} \) or \( 180^{\circ} \) (parallel or anti-parallel), the force will be zero.

This shows how crucial the direction and angle are in determining the force on a moving charge.

Electron Dynamics in Magnetic Fields

Electrons, like protons, are also affected by magnetic fields but behave slightly differently due to their negative charge. When an electron moves through the same magnetic field, it experiences the same magnitude of force as a proton (since they have the same charge magnitude), but the direction of the force is opposite.
This difference arises from the negative charge of electrons, which flips the direction of the magnetic force acting on them. For an electron traveling under the same conditions as a proton, it would encounter the same force, \( 3.70 \times 10^{-16} \mathrm{N} \), but in the reverse direction.

• The magnitude of the force remains contingent on the same factors: velocity, charge size, magnetic field strength, and the angle of entry.
• Understanding electron dynamics in magnetic fields is key in various applications, such as designing electronic components and understanding natural phenomena like auroras.

Overall, the dynamics of both electrons and protons help in comprehending complex magnetic interactions.

Applying the Right-Hand Rule

The right-hand rule is a helpful tool used to determine the direction of the force exerted on a moving charge in a magnetic field. It's a simple mnemonic that turns your hand into a guide.
Here's how it works:

• Point your thumb in the direction of the particle's velocity (motion).
• Point your fingers in the direction of the magnetic field lines.
• Your palm will then naturally face the direction of the force exerted on a positive charge like a proton.

For negative charges like electrons, the force direction will be opposite to that indicated by your palm.
Using the right-hand rule helps visualize the path a charged particle will take as it moves through a magnetic field, allowing us to predict and explain the deflection or trajectory changes observed in practice. Its hands-on approach makes it a valuable technique in both educational contexts and practical applications.