Problem 35 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

(a) Electron speed at point B is 9.53 × 10^5 m/s. (b) Proton speed at point B is 0 m/s.

Step 1: Understanding the Electric Field Effect

The electric field is directed towards the west, and since the electron is negatively charged, it will be accelerated towards the east. The force on the electron can be calculated using \( F = qE \), where \( q \) is the charge of the electron \( (q = -1.6 \times 10^{-19} \, C) \) and \( E \) is the electric field strength \((1.50 \, N/C)\).

Step 2: Calculating the Acceleration of the Electron

The acceleration \( a \) of the electron is given by \( a = \frac{F}{m} \), where \( F \) is the force found in Step 1, and \( m \) is the mass of the electron \((9.11 \times 10^{-31} \, kg)\). Substitute the values to find \( a \).

Step 3: Using Kinematics to Find Final Velocity of Electron

Since we know the initial velocity \( v_i = 4.50 \times 10^5 \, m/s \), we can use the kinematic equation \( v_f^2 = v_i^2 + 2ax \) to find the final velocity \( v_f \) of the electron, where \( x = 0.375 \, m \) is the displacement.

Step 4: Understanding the Electric Field Effect for Proton

For a proton, which is positively charged, the electric field and the force are in the same direction. Therefore, the proton will be decelerated as it moves eastwards. The force on the proton \( F = qE \), where \( q = +1.6 \times 10^{-19} \, C \).

Step 5: Calculating the Acceleration of the Proton

The acceleration \( a \) is calculated similarly as in step 2 for the proton, using its mass \((1.67 \times 10^{-27} \, kg)\).

Step 6: Using Kinematics to Find Final Velocity of Proton

Apply the kinematic equation \( v_f^2 = v_i^2 + 2ax \) again with the initial velocity of the proton \( v_i = 1.90 \times 10^4 \, m/s \) and calculate the final velocity \( v_f \), using the displacement \( x = 0.375 \, m \).

Key Concepts

Electron Motion

Electrons are tiny particles that carry a negative charge. When they enter an electric field, they experience a force due to their charge. According to the equation \( F = qE \) where \( q \) is the charge of the electron (\( -1.6 \times 10^{-19} \, C \)) and \( E \) is the electric field strength, we can calculate this force.
Since the electric field in our problem is directed towards the west, and considering an electron is moving east, it will be accelerated in the direction of its motion because it counters the electric field. This is similar to how a tailwind helps an airplane to speed up. Using the force calculated, we can proceed to calculate the electron's acceleration which will be used to determine its velocity at point B.

Proton Motion

Protons, unlike electrons, carry a positive charge. This fundamental difference leads to a different interaction with the electric field. The electric field exerts a force on a proton in the same direction as the field. Hence, a proton moving east in an electric field pointed west experiences deceleration.
Unlike the electron which was accelerated, the proton is slowed down as it fights against the electric field, much like a headwind slowing an aircraft. We use the same force equation \( F = qE \), with the charge of a proton being \(+1.6 \times 10^{-19} \, C\), to determine the force affecting it. Using this, one can calculate the negative acceleration that the proton undergoes as it moves from point A to B.

Kinematic Equations

Kinematic equations are mathematical tools that allow us to predict the motion of an object based on certain known variables. In the context of our electron and proton in motion, we use these equations to find their final velocities. One popular equation is \( v_f^2 = v_i^2 + 2ax \).
This kinematic equation relates the initial velocity (\( v_i \)), the final velocity (\( v_f \)), the acceleration (\( a \)), and the displacement (\( x \)).
For both the electron and proton, we know their initial velocities as they start moving at point A, and we need to calculate the final velocities at point B. By substituting the values (including the distance between points A and B: 0.375 m), we can solve for their respective final velocities.

Acceleration Calculation

Acceleration is a vital concept when it comes to predicting motion in physics. It is crucial when an object's speed is changing due to external forces, such as an electric field in our case. We calculate acceleration using the equation \( a = \frac{F}{m} \).
For an electron, use its mass: \( 9.11 \times 10^{-31} \, kg \), and for a proton, \( 1.67 \times 10^{-27} \, kg \). With the force already calculated (through \( F = qE \)), we substitute these into the acceleration formula for both the electron and proton. The resulting acceleration provides the change in speed over time as the particle moves from A to B. Understanding the signs of acceleration is crucial: a positive value indicates speeding up, while a negative value suggests slowing down.