Question

(a) An electron is to be accelerated from 3.00 \(\times 10^6\) m\(/\)s to 8.00 \(\times 10^6\) m\(/\)s. Through what potential difference must the electron pass to accomplish this? (b) Through what potential difference must the electron pass if it is to be slowed from 8.00 \(\times 10^6\) m\(/\)s to a halt?

Short answer

(a) 183.75 V, (b) 181.5 V.

Step-by-step solution

Understand the Energy Change Required

The energy change associated with an electron moving between two points at different speeds is captured by its change in kinetic energy. The kinetic energy \( K \) of an electron is given by \( K = \frac{1}{2}mv^2 \) where \( m \) is the mass of the electron and \( v \) is its velocity.

Calculate Initial and Final Kinetic Energies (Part a)

For the electron initially moving at 3.00 \( \times 10^6 \) m/s, the initial kinetic energy \( K_i \) is \( \frac{1}{2} \times 9.11 \times 10^{-31} \times (3.00 \times 10^6)^2 \).For the final velocity of 8.00 \( \times 10^6 \) m/s, the final kinetic energy \( K_f \) is \( \frac{1}{2} \times 9.11 \times 10^{-31} \times (8.00 \times 10^6)^2 \).

Compute Change in Kinetic Energy (Part a)

Calculate the change in kinetic energy \( \Delta K \) using the formula:\(\Delta K = K_f - K_i.\) Substitute the expressions from Step 2 to get \( \Delta K \).

Relate Kinetic Energy Change to Potential Difference (Part a)

The potential difference \( V \) needed to cause this change in kinetic energy is found using:\(\Delta K = e \Delta V,\)where \( e \) is the charge of the electron \( 1.60 \times 10^{-19} \) C. Solve for \( \Delta V \) using \( \Delta V = \frac{\Delta K}{e} \).

Initial and Final Kinetic Energies (Part b)

In part (b), calculate the initial kinetic energy \( K_i \) using the speed 8.00 \( \times 10^6 \) m/s.Since the final speed is zero, \( K_f = 0 \).

Compute Change in Kinetic Energy (Part b)

The change in kinetic energy \( \Delta K \) is given by:\(\Delta K = 0 - K_i.\)Calculate \( \Delta K \) using the \( K_i \) found in Step 5.

Relate Kinetic Energy Change to Potential Difference (Part b)

Again use the expression \( \Delta K = e \Delta V \) to find \( \Delta V \) in this scenario:\(\Delta V = \frac{\Delta K}{e}.\)Calculate \( \Delta V \) using the result for \( \Delta K \) from Step 6.

Key concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. For a moving body, like an electron, the kinetic energy (\( K \)) is given by the formula \( K = \frac{1}{2}mv^2 \).
This formula highlights two key factors that affect the kinetic energy: the mass (\( m \)) and the velocity (\( v \)) of the object.
For an electron, its mass is roughly \( 9.11 \times 10^{-31} \) kg.

• When the velocity of the electron changes, its kinetic energy changes as well.
• The greater the speed of the electron, the more kinetic energy it possesses.
• The change in kinetic energy can be calculated by finding the difference between the final kinetic energy and the initial kinetic energy.

This change in kinetic energy is useful when determining how much energy must be added or taken away to change the electron's speed.

Potential Difference

Potential difference, often referred to as voltage (\( V \)), is the difference in electric potential energy per unit charge between two points in an electric field.
It's a crucial concept when discussing the movement of electrons and other charged particles through different mediums.

• The potential difference is related to the energy needed to accelerate or decelerate a charge.
• In the context of electron acceleration, the potential difference that an electron passes through is directly related to its change in kinetic energy.
• The relationship between kinetic energy change and potential difference is expressed by the formula \( \Delta K = e \Delta V \), where \( e \) is the electron charge.

Thus, knowing the change in kinetic energy allows us to determine the potential difference necessary for that change.

Electron Charge

The electron charge (\( e \)) is a fundamental property of electrons that quantifies their electric charge. The value of the electron charge is \( 1.60 \times 10^{-19} \) coulombs, denoting it as a negatively charged particle.

• This charge is used in calculations relating to force, energy, and potential difference when dealing with electrons.
• It's a constant that allows us to relate the potential difference to the change in kinetic energy of an electron.
• Knowing the electron charge is essential for any calculation in electromagnetism that involves moving electrons.

The presence of a consistent charge value makes it possible to solve various physics problems involving electronic motion and acceleration.

Electron Mass

Electron mass is another intrinsic property of electrons, with a value of approximately \( 9.11 \times 10^{-31} \) kilograms.
This tiny mass plays a significant role in physics calculations, particularly when using the kinetic energy formula given its dependence on velocity.

• The low mass of electrons means they can achieve very high speeds with relatively low energy input.
• When determining the kinetic energy of an electron, the mass is necessary to apply the formula \( K = \frac{1}{2}mv^2 \).
• In addition to kinetic energy calculations, electron mass is important for understanding the behavior of electrons in fields and potentials.

Understanding the electron mass is vital for predicting how electrons will respond to forces and fields, essential for physics and electronic engineering.