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The problem describes a variation of Thomson's experiment, asking to calculate the velocity of cathode rays based on several given parameters. Let’s break it down step by step.

### Given:
- The radius of the coils: \( R = 27 \, \text{cm} = 0.27 \, \text{m} \)
- Number of turns in the coil: \( N = 234 \)
- The table provides values for the radius of the path \( r \) of the cathode rays in millimeters and the current \( I \) in amperes:
  - \( r = 66 \, \text{mm}, 58 \, \text{mm}, 53 \, \text{mm}, 49 \, \text{mm}, 44 \, \text{mm} \)
  - \( I = 1.6 \, \text{A}, 1.8 \, \text{A}, 2.0 \, \text{A}, 2.2 \, \text{A}, 2.4 \, \text{A} \)

### Formula:
Thomson's experiment relates the magnetic field \( B \) produced by the coil, the radius of the path \( r \), and the velocity \( v \) of the electron by the following relationship:
\[
B = \frac{{\mu_0 N I}}{{2R}}
\]
where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A} \) is the permeability of free space, \( N \) is the number of turns, \( I \) is the current, and \( R \) is the radius of the coil.

The magnetic force on a charged particle moving in a circular path is balanced by the centripetal force, which gives:
\[
v = \frac{{eBr}}{{m}}
\]
where \( e \) is the charge of the electron (\( e = 1.6 \times 10^{-19} \, \text{C} \)) and \( m \) is the mass of the electron (\( m = 9.11 \times 10^{-31} \, \text{kg} \)).

Now, I will calculate the magnetic field \( B \) for each current, and then use the equation to solve for the velocity \( v \). Let me proceed with the calculations.The calculated velocities of the cathode rays corresponding to the given data are:

- For \( r = 66 \, \text{mm} \) and \( I = 1.6 \, \text{A} \), \( v \approx 10,099,444 \, \text{m/s} \)
- For \( r = 58 \, \text{mm} \) and \( I = 1.8 \, \text{A} \), \( v \approx 9,984,678 \, \text{m/s} \)
- For \( r = 53 \, \text{mm} \) and \( I = 2.0 \, \text{A} \), \( v \approx 10,137,700 \, \text{m/s} \)
- For \( r = 49 \, \text{mm} \) and \( I = 2.2 \, \text{A} \), \( v \approx 10,309,850 \, \text{m/s} \)
- For \( r = 44 \, \text{mm} \) and \( I = 2.4 \, \text{A} \), \( v \approx 10,099,444 \, \text{m/s} \)

The velocities are quite consistent, with an average velocity close to \( 10,100,000 \, \text{m/s} \).

Do you need further details or have any questions?

Here are 5 related questions:
1. How does the magnetic field strength affect the velocity of cathode rays?
2. How would the results change if the coil's radius was increased?
3. What are the main differences between Thomson's experiment and the modern versions of measuring electron velocity?
4. How does increasing the number of turns in the coil affect the velocity of the electrons?
5. Why does the velocity stay relatively constant despite changes in current and radius?

**Tip**: When conducting experiments involving magnetic fields, ensure the uniformity of the field across the entire region where the electron path is observed to minimize errors in velocity calculations.

Se repite el experimento de Thomson similar al del laboratorio, donde el radio de las bobinas es 27 cm y el número de espiras es 234. Use la totalidad de la información y calcule el mejor valor para la velocidad de los rayos catódicos.

In this problem, we calculate the velocity of cathode rays using Thomson's experiment with a coil radius of 27 cm and 234 turns. The given data includes the radius of the electron path and current values, leading to an average velocity calculation of approximately 10,100,000 m/s.

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