Problem 28 - University Physics with Modern Physics

Question

(a) What is the speed of a beam of electrons when the simultaneous influence of an electric field of 1.56 \(\times\) 10\(^4\) V/m and a magnetic field of 4.62 \(\times\) 10\(^{-3}\) T, with both fields normal to the beam and to each other, produces no deflection of the electrons? (b) In a diagram, show the relative orientation of the vectors \(\vec{v}\), \(\overrightarrow{E}\), and \(\overrightarrow{B}\). (c) When the electric field is removed, what is the radius of the electron orbit? What is the period of the orbit?

Step-by-Step Solution

Answer

The speed is approximately 3.38 × 10^6 m/s. The orbit's radius is about 4.17 × 10^-2 m, and the period is around 7.76 × 10^-8 seconds.

Step 1: Identify Forces on the Electron

When an electron moves through both electric \(\overrightarrow{E}\) and magnetic \(\overrightarrow{B}\) fields, it experiences forces: the electric force \(\overrightarrow{F_E} = e\overrightarrow{E}\), and the magnetic force \(\overrightarrow{F_B} = e\overrightarrow{v} \times \overrightarrow{B}\). For no deflection, \(\overrightarrow{F_E} = \overrightarrow{F_B}\).

Step 2: Calculate the Velocity for No Deflection

Since the forces are equal and opposite, for no deflection the magnitude of these forces are equal: \(eE = evB\), leading to the electron speed given by \(v = \frac{E}{B}\). Substitute \(E = 1.56 \times 10^4\) V/m and \(B = 4.62 \times 10^{-3}\) T into the equation.

Step 3: Compute the Electron Speed

Use the equation \(v = \frac{1.56 \times 10^4}{4.62 \times 10^{-3}}\) to calculate the speed of the electrons. Calculate \(v\) to find the answer in m/s.

Step 4: Diagram the Vector Orientation

In the scenario of no deflection, the electric field \(\overrightarrow{E}\), the magnetic field \(\overrightarrow{B}\), and the velocity \(\vec{v}\) are mutually perpendicular. Draw \(\overrightarrow{E}\) and \(\overrightarrow{B}\) perpendicular to each other on a plane, and \(\overrightarrow{v}\) perpendicular to that plane.

Step 5: Determine the Radius of the Electron Orbit

Once the electric field is removed, the electron will only be influenced by the magnetic field. The radius of the orbit \(r\) in a magnetic field is given by \(r = \frac{mv}{eB}\), where \(m\) is the electron mass and \(e\) is the charge of an electron. Use \(m = 9.11 \times 10^{-31}\) kg, \(e = 1.60 \times 10^{-19}\) C, and the previously calculated \(v\).

Step 6: Calculate the Period of the Orbit

The period \(T\) of an electron's orbit in a magnetic field is given by \(T = \frac{2\pi r}{v}\). Use the calculated radius \(r\) and electron speed \(v\) to determine \(T\).

Key Concepts

Electric Force

When examining the motion of electrons in the presence of electric fields, we encounter the concept of electric force. This force is the result of the electric field \(\overrightarrow{E}\) acting on a charged particle such as an electron. The formula for calculating the electric force \(\overrightarrow{F_E}\) is given by \(\overrightarrow{F_E} = e\overrightarrow{E}\), where \(e\) represents the elementary charge, approximately \(-1.60 \times 10^{-19}\) C, characteristic of electrons.

This force contributes to the electron's acceleration in the direction of the field if unopposed. However, in our scenario, the presence of a magnetic field exerts a simultaneous magnetic force, potentially countering the electric force to result in no net deflection of the electron's path. Understanding this balance is crucial in identifying conditions for a stable electron trajectory in combined electric and magnetic fields.

Magnetic Force

Magnetic force is another critical factor in the movement of electrons within a magnetic field. Unlike electric force, magnetic force depends on both the velocity of the electron and the magnetic field. The expression for magnetic force \(\overrightarrow{F_B}\) acting on a charged particle moving through a magnetic field \(\overrightarrow{B}\) is given by: \(\overrightarrow{F_B} = e\overrightarrow{v} \times \overrightarrow{B}\).

Notice the cross product, indicating that the magnetic force is perpendicular to both the velocity \(\overrightarrow{v}\) and the magnetic field vector \(\overrightarrow{B}\). It is this unique orientation that can lead to circular paths when no electric force counters it. In the exercise, the condition that there's no deflection means that the electron's perceived path remains straight in the combined fields, achievable when the magnitudes of the electric and magnetic forces are equal, thereby balancing each other out. This is expressed mathematically as \(eE = evB\), leading to a relation for the electron's velocity.

Electron Orbit Radius

Once the electric field is removed, only the magnetic field remains to influence the electron. Under these circumstances, the electron begins to travel in a circular orbit. The radius of this orbit \(r\) in the magnetic field can be determined using the formula: \(r = \frac{mv}{eB}\), where \(m\) is the electron's mass (approximately \(9.11 \times 10^{-31}\) kg), \(v\) is the velocity previously calculated, and \(e\) is the elementary charge.

The magnetic force acts as the centripetal force, maintaining the electron's circular motion by pulling it inward. Calculating \(r\) provides insight into the physical path the electron will traverse under these conditions. This understanding is particularly useful in areas like designing cyclotrons and understanding particle tracks within magnetic fields.

Vector Orientation in Fields

Understanding the orientation of vectors is vital when visualizing the scenario described in the exercise. Here, the electric field vector \(\overrightarrow{E}\), magnetic field vector \(\overrightarrow{B}\), and electron velocity \(\vec{v}\) must all be perpendicular to one another. This mutual perpendicularity ensures that the forces can balance out, causing no net deflection of the electron's trajectory in the electric and magnetic fields.

To illustrate: if the electric field is aligned along the x-axis, the magnetic field might be along the y-axis, making the electron's velocity vector aligned along the z-axis. Each vector's perpendicular arrangement ensures the forces resulting from each field can counteract the others, allowing electrons to maintain a straight path without deflection. In physics, such vector orientations help solve complex problems, clarifying how these entities interact in space.