Calculating the Velocity of an Electron Accelerated by a Voltage

Introduction

Understanding the velocity of an electron accelerated by a specific voltage is a fundamental concept in physics, important for both academic and practical applications. In this article, we will walk through the mathematical calculations necessary to determine the velocity of an electron when subjected to a 1000-volt potential difference.

Key Concepts

The relationship between the velocity of an electron and the voltage through which it is accelerated involves the concepts of potential energy and kinetic energy. The formula to determine the velocity is given by:

v 2qV/m

Where:

q is the charge of the electron (1.6 × 10-19 C) V is the potential difference in volts (1000 volts) m is the mass of the electron (9.1 × 10-31 kg)

Calculation Using a Non-Relativistic Approximation

Let's work through the calculation with a non-relativistic approximation. This is valid for small accelerations where relativistic effects are negligible.

q  1.6 times; 10-19 C
m  9.1 times; 10-31 kg
V  1000 volts
v  2 times; q times; V / m
v  2 times; 1.6 times; 10-19 times; 1000 / 9.1 times; 10-31
v  3.2 times; 1013 m/s

Therefore, the velocity of the electron is approximately 1.8752 times; 107 m/s or simply 1.9 times; 107 m/s.

Relativistic Calculation

For higher voltages, relativistic effects must be considered. The relativistic expression for kinetic energy (KE) is:

KE  (gamma - 1)mc^2

Where theta is the total energy and gamma is the Lorentz factor, given by:

gamma 1 (KE/mc^2)

KE  2 times; 1000 eV  2 keV  2000 eV

Given that the rest mass energy of an electron is:

mc^2 511 keV

we can solve for gamma:

gamma  1   2000/511 keV

Substituting and simplifying:

gamma 1 3.907 4.907

Since gamma is quite large, it confirms that relativistic effects are significant. The velocity can be found using the non-relativistic approximation as a good estimate, but to be precise, we would need to use the full relativistic formula.

Alternative Methods and Approximations

Another method to find the velocity is through the conversion of potential energy directly into kinetic energy:

1 eV 1.602 x 10-19 J

At 1000 volts, the electron gains:

1000 eV 1.602 x 10-16 J

Using the kinetic energy formula:

1.602 times; 10-16  0.5 times; 9.1 times; 10-31 times; v^2
v^2  (2 times; 1.602 times; 10-16 / 9.1 times; 10-31)
v  2 x 1.602 x 10^-16 / 9.1 x 10^-31
v  3.5164 times; 10^14 m/s

This results in:

v 1.8752 times; 107 m/s

Which confirms our previous calculations.

Conclusion

Understanding the velocity of an electron accelerated by a voltage is a crucial concept in physics. Whether using a non-relativistic or relativistic approach, the calculations yield a velocity of approximately 1.8752 times; 107 m/s. This value can be used in various applications ranging from practical experiments to theoretical calculations.