To determine the charge-to-mass ratio (e/m) of the electrons in the given scenario, we can use the information provided about the electron gun, the magnetic field, and the radius of the circular path of the electrons. Let's break this down step by step.
Understanding the Basics
The charge-to-mass ratio (e/m) is a fundamental property of charged particles, such as electrons. It can be calculated using the formula:
e/m = (2V) / (B²r²)
Where:
- V = potential difference (in volts)
- B = magnetic field strength (in teslas)
- r = radius of the circular path (in meters)
Gathering the Data
From the problem statement, we have:
- Potential difference, V = 100 V
- Magnetic field strength, B = 2.83 × 10-4 T
- Radius of the circular path, r = 12.0 cm = 0.12 m
Plugging in the Values
Now, we can substitute these values into the formula to find e/m:
e/m = (2 × 100 V) / ((2.83 × 10-4 T)² × (0.12 m)²)
Calculating Step-by-Step
First, calculate the denominator:
B² = (2.83 × 10-4)² = 8.0089 × 10-8 T²
r² = (0.12)² = 0.0144 m²
Now, multiply B² by r²:
B²r² = (8.0089 × 10-8 T²) × (0.0144 m²) = 1.1533 × 10-9 T²·m²
Next, calculate the numerator:
2V = 2 × 100 V = 200 V
Now, we can find e/m:
e/m = 200 V / (1.1533 × 10-9 T²·m²)
Calculating this gives:
e/m ≈ 1.733 × 1011 C/kg
Final Result
Thus, the charge-to-mass ratio (e/m) of the electrons in this experiment is approximately 1.733 × 1011 C/kg. This value is consistent with the known charge-to-mass ratio of electrons, which is about 1.76 × 1011 C/kg, confirming the accuracy of our calculations.
