Problem 6 - University Physics with Modern Physics

Question

An electron moves at \(2.50 \times 10^{6} \mathrm{m} / \mathrm{s}\) through a region in which there is a magnetic field of unspecified direction and magnitude \(7.40 \times 10^{-2} \mathrm{T}\) (a) What are the largest and smallest possible magnitudes of the acceleration of the electron due to the magnetic field? (b) If the actual acceleration of the electron is one-fourth of the largest magnitude in part (a), what is the angle between the electron velocity and the magnetic field?

Step-by-Step Solution

Answer

The largest acceleration is when velocity is perpendicular; smallest is zero. For part (b), solve for \(\theta\) using \(\sin\theta = \frac{1}{4}\).

Step 1: Understand the Scenario

We have an electron moving with a velocity of \(2.50 \times 10^6 \ \text{m/s}\) in a magnetic field with a magnitude of \(7.40 \times 10^{-2} \ \text{T}\). The Lorentz force due to the magnetic field acts perpendicularly to the velocity of the electron.

Step 2: Calculate the Maximum Force on the Electron

The force on a charged particle in a magnetic field is given by \(F = qvB\sin\theta\). The maximum force occurs when \(\sin\theta = 1\), i.e., the velocity is perpendicular to the magnetic field. Thus, the maximum force is \(F_{\text{max}} = qe \cdot v \cdot B = (1.6 \times 10^{-19} \ \text{C}) \cdot (2.50 \times 10^6 \ \text{m/s}) \cdot (7.40 \times 10^{-2} \ \text{T})\).

Step 3: Calculate the Maximum Acceleration

The acceleration can be determined from \(a = \frac{F}{m}\), where \(m\) is the mass of the electron \((9.11 \times 10^{-31} \ \text{kg})\). Thus, the maximum acceleration \(a_{\text{max}}\) is given by \(a_{\text{max}} = \frac{F_{\text{max}}}{m}\).

Step 4: Determine the Minimum Acceleration

The minimum acceleration is zero, which occurs when the velocity is parallel to the magnetic field (\(\theta = 0^\circ\) or \(180^\circ\)), resulting in \(F = 0\).

Step 5: Calculate the Actual Acceleration

We know that the actual acceleration is one-fourth of the maximum. If \(a' = \frac{1}{4} a_{\text{max}}\), we can use this relationship to find the value of the applied acceleration.

Step 6: Find the Angle Between Velocity and Magnetic Field

Using \( a' = \frac{e \cdot v \cdot B \cdot \sin\theta}{m}\), we rearrange to find \(\theta\): \( \sin\theta = \frac{a' \cdot m}{e \cdot v \cdot B} \). Solving for \(\theta\) gives us the angle between the electron's velocity and the magnetic field.

Key Concepts

Electron motion in magnetic field

When we talk about electrons moving in a magnetic field, we're looking at how they're influenced by the magnetic force, known as the Lorentz force. This force is perpendicular to both the electron's velocity and the direction of the magnetic field.

The equation governing this interaction is the Lorentz force formula:

• \( F = qvB \sin \theta \)\( \)

Here,

• \( F \) is the force on the electron,
• \( q \) is the charge of the electron,
• \( v \) is the velocity of the electron,
• \( B \) is the magnetic field strength,
• \( \theta \) is the angle between the electron's velocity and the magnetic field direction.

For an electron, which has a negative charge, the direction of the force is opposite to that predicted by the standard right-hand rule. This perpendicular force causes the electron to travel in a circular or spiral path rather than a straight line. The motion of an electron in a magnetic field is a fundamental concept in physics, important for understanding technologies like CRT monitors and magnetic storage devices.

Maximum and minimum acceleration

The acceleration an electron experiences in a magnetic field is derived from the force acting on it. Since force and acceleration are directly related by Newton's second law (\( a = \frac{F}{m} \)), understanding the maximum and minimum possible forces allows us to determine the electron's acceleration limits.

**Maximum Acceleration:**

• This occurs when the force is at its maximum, which means the velocity of the electron is perpendicular to the magnetic field (\( \sin \theta = 1 \)).
• The maximum force is \( F_{\text{max}} = qvB \), and so maximum acceleration is \( a_{\text{max}} = \frac{qvB}{m} \).

**Minimum Acceleration:**

• Occurs when the velocity vector is either parallel or anti-parallel to the magnetic field (\( \theta = 0^\circ \) or \( 180^\circ \)), causing \( \sin \theta = 0 \).
• In this case, the force is zero, resulting in zero acceleration.

Thus, by controlling this angle \( \theta \), we can manipulate the acceleration of electrons in devices such as particle accelerators.

Angle between velocity and magnetic field

The angle \( \theta \) between an electron's velocity and the magnetic field is crucial because it affects both the direction and magnitude of the force, as we saw previously with the Lorentz force equation.

To find \( \theta \), if we're given that the electron's acceleration is a fraction of its maximum possible value, we can use:

• \( a' = \frac{1}{4} a_{\text{max}} \)

For a known \( a' \) based on this condition, rearranging the Lorentz force equation helps solve for \( \theta \):

• \( \sin \theta = \frac{a' \cdot m}{q \cdot v \cdot B} \)

Calculating \( \sin \theta \) provides \( \theta \), but it's important to ensure it falls within possible values, which is between \( 0^\circ \) and \( 90^\circ \). This assessment of \( \theta \) helps in understanding how charged particles can be directed and manipulated in practical applications like magnetic resonance imaging (MRI). By controlling \( \theta \), devices can achieve the desired particle path and performance efficiently.