A particle with charge +e has a total energy of 0.638 MeV when it is moving at 0.600 c. If this

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A particle with charge +e has a total energy of 0.638 MeV...

A particle with charge +e has a total energy of $0.638 \mathrm{MeV}$ when it is moving at $0.600 \mathrm{c}$. If this particle then enters a linear accelerator, what is its speed after it has been accelerated through a 2.6 MV potential difference?

Key Concepts

Energy Conservation in Relativistic Systems

When a charged particle is accelerated, the energy gained from the electric field adds to its kinetic energy. In a relativistic context, the conservation of energy requires accounting for both the particle's rest energy (mc²) and the relativistic kinetic energy. This principle allows one to relate the work done by the accelerating potential to the final speed of the particle by equating the total energy before and after acceleration.

Electric Potential Difference

A potential difference (voltage) across an accelerator is used to do work on charged particles. The work done is equal to the charge multiplied by the potential difference, providing an energy boost to the particle. This concept is central in determining the energy gained by a particle as it passes through regions of differing electric potential in accelerators.

Relativistic Kinetic Energy

In special relativity, the kinetic energy of a particle moving at speeds comparable to the speed of light is given by the expression (? - 1)mc², where ? (the Lorentz factor) accounts for time dilation and length contraction at high velocities. This formulation replaces the classical kinetic energy formula and is essential for accurately describing energy changes when accelerating particles to relativistic speeds.

Lorentz Factor

The Lorentz factor, defined as ? = 1/?(1 - (v/c)²), quantifies the effects of relativistic speeds on time, length, and energy. It plays a critical role in converting between rest energy and total energy, and is fundamental in calculating the increase in kinetic energy as a particle’s speed approaches the speed of light.

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