Electrons are accelerated through a potential difference of 750750 kV, so that their kinetic energy is 7.50×1057.50\$$times10^5 eV.(a) $$What is the ratio of the speed vv of an electron having this energy to the speed of light, cc?(b) What would the speed be if it were computed from the principles of classical mechanics?

Ch 37: Special Relativity

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 37: Special Relativity

Problem 36

Chapter 37, Problem 36

Electrons are accelerated through a potential difference of $750$ kV, so that their kinetic energy is $7.50 \times 10^{5}$ eV.
(a) What is the ratio of the speed $v$ of an electron having this energy to the speed of light, $c$ ?
(b) What would the speed be if it were computed from the principles of classical mechanics?

Verified step by step guidance

Step 1: Understand the relationship between kinetic energy and speed for an electron. The kinetic energy (KE) of an electron accelerated through a potential difference is given by the equation KE = eV, where e is the charge of the electron and V is the potential difference.

Step 2: For part (a), use the relativistic energy-momentum relation to find the speed of the electron. The relativistic kinetic energy is given by KE = γmc² - mc², where γ is the Lorentz factor, m is the rest mass of the electron, and c is the speed of light. Solve for γ using the given kinetic energy.

Step 3: Calculate the Lorentz factor γ using the equation γ = 1 / sqrt(1 - (v²/c²)). Rearrange this equation to solve for the speed v of the electron in terms of c.

Step 4: For part (b), use classical mechanics to find the speed of the electron. The classical kinetic energy is given by KE = 0.5mv². Rearrange this equation to solve for v using the given kinetic energy.

Step 5: Compare the results from the relativistic and classical calculations to understand the difference in speeds. Note that relativistic effects become significant at speeds close to the speed of light, which is why the classical calculation may differ significantly from the relativistic one.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Potential Difference and Kinetic Energy

Potential difference, measured in volts, is the work done per unit charge to move a charge between two points. When electrons are accelerated through a potential difference, they gain kinetic energy equal to the charge multiplied by the potential difference. In this case, the kinetic energy of the electrons is given as 7.50 * 10^5 eV, which is derived from the potential difference of 750 kV.

Relativistic Speed and Speed of Light

The speed of light, denoted as c, is approximately 3.00 * 10^8 m/s and represents the maximum speed at which information or matter can travel. When particles like electrons are accelerated to high energies, their speeds approach c, requiring relativistic physics to accurately describe their motion. The ratio of the electron's speed to c helps determine how relativistic effects influence the electron's behavior.

Classical Mechanics vs. Relativistic Mechanics

Classical mechanics, based on Newton's laws, assumes that speeds are much less than the speed of light, allowing for straightforward calculations of velocity using kinetic energy. However, at high speeds, relativistic mechanics must be used, as it accounts for the increase in mass and energy effects at velocities approaching c. The question contrasts these two approaches to highlight the differences in calculated speeds.

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