e/m Calculator (Charge-to-Mass Ratio)
This calculator determines the charge-to-mass ratio (e/m) of an electron based on the principles of J.J. Thomson's famous 1897 experiment. By inputting the accelerating voltage, magnetic field strength, and the radius of the electron's path, you can compute this fundamental physical constant.
Understanding the Charge-to-Mass Ratio (e/m)
The charge-to-mass ratio (e/m) is a fundamental property of a charged particle. It represents the particle's electric charge divided by its mass. The discovery and measurement of this ratio for the electron were pivotal moments in physics, proving the existence of subatomic particles and paving the way for the development of atomic models. A high e/m ratio indicates that a particle is very light for its charge, which is characteristic of the electron.
How is the e/m Ratio Measured?
In a typical experimental setup (like a cathode ray tube):
- Electrons are emitted from a cathode and accelerated through a known potential difference (Voltage, V). This gives them a specific kinetic energy.
- These accelerated electrons then enter a region with a uniform magnetic field (B), applied perpendicular to their direction of travel.
- The magnetic force acts as a centripetal force, causing the electrons to move in a circular path of a specific radius (r).
By measuring the accelerating voltage (V), the magnetic field strength (B), and the radius of the circular path (r), we can calculate the e/m ratio.
The Formula for Calculating e/m
The relationship between these variables is derived by equating the kinetic energy gained by the electron to the work done by the electric field, and the magnetic force to the centripetal force. The resulting formula is:
Where:
- e/m is the charge-to-mass ratio in Coulombs per kilogram (C/kg).
- V is the accelerating voltage in Volts (V).
- B is the magnetic field strength in Tesla (T).
- r is the radius of the electron's circular path in meters (m).
Calculate the e/m Ratio
Worked Example
Let's say in an experiment, you use an accelerating voltage of 400 V and a magnetic field of 0.0012 T. You observe that the electron beam curves into a path with a radius of 0.11 meters.
- V = 400 V
- B = 0.0012 T
- r = 0.11 m
Using the formula:
e/m = (2 * 400) / (0.0012² * 0.11²)
e/m = 800 / (0.00000144 * 0.0121)
e/m = 800 / (1.7424e-8)
e/m ≈ 1.765 x 10¹¹ C/kg
This calculated value is very close to the accepted theoretical value.
Accepted Value and Experimental Error
The currently accepted value for the electron's charge-to-mass ratio is approximately 1.759 × 10¹¹ C/kg. Discrepancies between the calculated value and the accepted value are common in real-world experiments due to measurement uncertainties, non-uniformity in the magnetic field, and other environmental factors.