Problem 46 - College Physics

Question

\(\bullet\) \(\bullet\) (a) An electron is moving east in a uniform electric field of 1.50 \(\mathrm{N} / \mathrm{C}\) directed to the west. At point \(A\) , the velocity of the electron is \(4.50 \times 10^{5} \mathrm{m} / \mathrm{s}\) toward the east. What is the speed of the electron when it reaches point \(B, 0.375 \mathrm{m}\) east of point \(A\) ? (b) A proton is moving in the uniform electric field of part (a). At point \(A\) , the velocity of the proton is \(1.90 \times 10^{4} \mathrm{m} / \mathrm{s},\) east. What is the speed of the proton at point \(B\) ?

Step-by-Step Solution

Answer

The electron speeds up to a higher velocity at point B; the proton slows down.

Step 1: Understand the Forces

The electric force acting on a charge is given by the formula: \( F = qE \), where \( q \) is the charge and \( E \) is the electric field strength. For an electron, \( q = -1.6 \times 10^{-19} \mathrm{C} \), and for a proton, \( q = +1.6 \times 10^{-19} \mathrm{C} \). The electric field is \( 1.50 \, \mathrm{N/C} \) to the west, which means a force to the east for an electron and the opposite for a proton.

Step 2: Calculate Acceleration

Using the formula \( F = ma \) and substituting \( F = qE \), we can find the acceleration: \( a = \frac{qE}{m} \). For an electron, \( m = 9.11 \times 10^{-31} \, \mathrm{kg} \), and for a proton, \( m = 1.67 \times 10^{-27} \, \mathrm{kg} \). Calculate \( a_e \) for the electron and \( a_p \) for the proton.

Step 3: Apply the Kinematic Equation

Use the equation \( v^2 = u^2 + 2as \) to find the final velocity \( v \) at point \( B \), where \( u \) is the initial velocity, \( a \) is the acceleration (from Step 2), and \( s = 0.375 \, \mathrm{m} \) is the distance.

Step 4: Solve for Electron Velocity

For the electron, substitute \( u = 4.50 \times 10^5 \, \mathrm{m/s} \) and the respective \( a_e \) from Step 2 into the kinematic equation. Solve for \( v_e \).

Step 5: Solve for Proton Velocity

For the proton, substitute \( u = 1.90 \times 10^4 \, \mathrm{m/s} \) and \( a_p \) from Step 2 into the kinematic equation. Solve for \( v_p \).

Step 6: Conclusion

The electron will increase in speed due to the force acting in the direction of its velocity, while the proton will slow down. Use the numerical values of \( a_e \) and \( a_p \) to calculate the exact speeds at point \( B \).

Key Concepts

Electron Motion

Electrons are negatively charged particles that move very quickly, especially in the presence of an electric field. When an electron moves through an electric field, it experiences a force due to its charge. This electric force is given by the formula:

• \( F = qE \),

where \( F \) is the force, \( q \) is the charge of the electron, and \( E \) is the electric field strength.
In our scenario, the electric field is directed to the west, but since the electron itself has a negative charge, the force acting on it is directed towards the east. This means that when an electron is moving east through the electric field, it speeds up because the direction of the force complements the direction of the electron's initial motion.

When calculating the electron's acceleration in the electric field, we use the equation:

• \( a = \frac{qE}{m} \),

where \( m \) is the mass of an electron. Substituting values gives us the acceleration which can then be used with kinematic equations to find changes in velocity over a specified distance.

Proton Motion

Protons, unlike electrons, have a positive charge. They are the atomic nucleus constituents and are much heavier than electrons. When placed in an electric field, protons experience a force given by:

• \( F = qE \),

where \( q \) for a proton is a positive charge. In the same electric field directed west, protons will experience a force directed to the west as well.
Since the protons are initially moving east, the force acts against the proton's motion, leading to a deceleration.

To calculate this change in speed when moving from point \( A \) to point \( B \) in the field, we apply the formula for acceleration:

• \( a = \frac{qE}{m} \),

using the mass of the proton, which is significantly larger than that of an electron. This heavier mass results in a smaller acceleration compared to that of the electron.

Kinematic Equations

Kinematic equations are essential to solving motion problems in physics, especially when dealing with constant forces like gravity or electric fields.
These equations allow us to relate the initial and final velocities of an object, its acceleration, and the distance it has traveled. A common kinematic equation used is:

• \( v^2 = u^2 + 2as \),

where:

• \( v \) is the final velocity,
• \( u \) is the initial velocity,
• \( a \) is the acceleration,
• \( s \) is the distance covered.

For the electron and proton moving from point \( A \) to point \( B \), we apply this equation to determine the final velocities. The direction and magnitude of acceleration play crucial roles in increasing or decreasing the speed, depending on whether the object is moving with or against the electric field's force.

Electric Force

Electric force is a fundamental concept in electrostatics. It arises due to the interaction between charged particles and electric fields.
In our exercise, the electric force acts differently on electrons and protons due to their opposite charges. For any charged particle in an electric field, the force can be expressed by the equation:

• \( F = qE \).

Here, \( F \) represents the force exerted on the charge, \( q \) is the charge value, and \( E \) is the electric field's intensity. This force directly influences the acceleration of charged particles as per Newton's second law, \( F = ma \), where \( m \) is the mass and \( a \) is the acceleration.

This interplay affects how electrons and protons move when subjected to an electric field. Electrons feel a force in the opposite direction to the field due to their negative charge, while protons feel a force in the same direction due to their positive charge. Thus, understanding electric force is key to predicting and calculating the motion of charged particles in fields.