Problem 9 - Physics

Question

The electrons in the beam of a television tube have a kinetic energy of \(2.40 \times 10^{-15} \mathrm{~J}\). Initially, the electrons move horizontally from west to east. The vertical component of the earth's magnetic field points down, toward the surface of the earth, and has a magnitude of \(2.00 \times 10^{-5} \mathrm{~T}\). (a) In what direction are the electrons deflected by this field component? (b) What is the acceleration of an electron in part (a)?

Step-by-Step Solution

Answer

(a) Electrons are deflected north. (b) Acceleration is \(2.56 \times 10^{14} \mathrm{~m/s^2}.\)

Step 1: Understand the problem

We need to determine the direction and magnitude of the deflection of electrons due to the Earth's magnetic field. Electrons move initially horizontally, while the Earth's magnetic field has a vertical component pointing downward. By applying the right-hand rule, we can find the direction of the force on a negative charge.

Step 2: Apply the Right-Hand Rule

The electrons are initially moving east, and the magnetic field points downward. Using the right-hand rule for negative charges (like electrons), point the fingers of your right hand in the direction of the electron velocity (east), and curl them toward the magnetic field direction (down). The thumb will point in the direction of the magnetic force, which is north.

Step 3: Calculate the Velocity of the Electrons

The kinetic energy (KE) is given by the equation \( KE = \frac{1}{2} mv^2 \), where \(m\) is the mass of an electron \( (9.11 \times 10^{-31} \mathrm{~kg}) \). Rearrange to solve for velocity:\[ v = \sqrt{\frac{2 \times KE}{m}} = \sqrt{\frac{2 \times 2.40 \times 10^{-15}}{9.11 \times 10^{-31}}} \approx 7.29 \times 10^7 \mathrm{~m/s}. \]

Step 4: Calculate the Magnetic Force

The force on a charged particle in a magnetic field is \( F = qvB \sin \theta \), where \( q \) is the charge of the electron \( (1.6 \times 10^{-19} \mathrm{~C}) \), and \( \theta = 90^\circ \) here as the motion is perpendicular to the magnetic field:\[ F = (1.6 \times 10^{-19}) \times (7.29 \times 10^7) \times (2.00 \times 10^{-5}) \approx 2.33 \times 10^{-16} \mathrm{~N}. \]

Step 5: Calculate the Acceleration

Use Newton's second law to find the acceleration, \( F = ma \). Thus:\[ a = \frac{F}{m} = \frac{2.33 \times 10^{-16}}{9.11 \times 10^{-31}} \approx 2.56 \times 10^{14} \mathrm{~m/s^2}. \]

Key Concepts

Earth's Magnetic Field

The Earth is not just a giant ball of rock. It has a magnetic field that surrounds it and affects charged particles, like electrons, when they pass through it. This magnetic field has two main components: vertical and horizontal. In the context of our problem, the vertical component points in a downward direction, which means it is directed towards the Earth's surface. The strength of the Earth's magnetic field can vary depending on location and other factors, but in our example, it has a magnitude of \(2.00 \times 10^{-5} \mathrm{~T}\).
This magnetic field plays a crucial role in determining the path of moving charged particles, such as electrons, by applying a force on them. When electrons move horizontally from west to east, the vertical magnetic field will exert a force on them, causing deflection. This interaction highlights the dynamic nature of Earth's magnetic field and its impact on electronics and other charged objects.

Kinetic Energy of Electrons

Electrons in motion possess kinetic energy, which is the energy due to their movement. The kinetic energy of an electron can be calculated using the formula:

• \( KE = \frac{1}{2} mv^2 \)

Here, \( m \) is the electron's mass \((9.11 \times 10^{-31} \mathrm{~kg})\) and \( v \) is its velocity.
From the problem, we know that the electrons have a kinetic energy of \(2.40 \times 10^{-15} \mathrm{~J}\). This information allows us to find their velocity by rearranging the kinetic energy equation to solve for \( v \):

• \( v = \sqrt{\frac{2 \times KE}{m}} \)

Substituting the given values, we find the electrons are moving at approximately \(7.29 \times 10^7 \mathrm{~m/s}\).
Understanding the kinetic energy helps in predicting how electrons will behave under the influence of an outside force, such as a magnetic field.

Right-Hand Rule

The right-hand rule is a valuable tool in physics used to determine the direction of force on charged particles in a magnetic field. To apply this rule, particularly for electrons or negative charges, follow these steps:

• Point your fingers in the direction of electron velocity (eastward in this case).
• Curl your fingers towards the direction of the magnetic field (downward in this scenario).

Your thumb will then point in the direction of the magnetic force acting on the electrons. As electrons are negatively charged, the actual force direction is opposite to the thumb's direction. Thus, in this situation, instead of south, they are deflected towards the north.
This simple yet effective technique provides a clear visual method to discern the effect of magnetic fields on charged particles, aiding in comprehending complex electromagnetic interactions.

Acceleration of Electrons

Once the force exerted by the magnetic field on the moving electrons is known, determining the acceleration is straightforward using Newton's second law of motion. This law states that force is equal to mass times acceleration:

• \( F = ma \)

Rearranging this equation to solve for acceleration \( a \), we have:

• \( a = \frac{F}{m} \)

From previous calculations, the magnetic force \( F \) acting on the electrons is \(2.33 \times 10^{-16} \mathrm{~N}\).
Using the mass of the electron \( (9.11 \times 10^{-31} \mathrm{~kg}) \), we can calculate:

• \( a \approx 2.56 \times 10^{14} \mathrm{~m/s^2} \)

This acceleration is a result of the electrons interacting with the Earth's magnetic field and aids in predicting their new trajectories under such forces.