Question

A tightly wound long solenoid carries a current of 1.5 A. An electron is executing uniform circular motion inside the solenoid with a time period of 75 ns. The number of turns per meter in the solenoid is ________________.

Hint:

The magnetic field in a solenoid is directly proportional to the current and the number of turns per unit length. Use the magnetic force on the electron to relate the field and the motion.

Answer 1

Step 1: Understand the Given Information

The problem provides the following information:

• The current in the solenoid is 1.5 A.
  
• An electron is executing uniform circular motion inside the solenoid with a time period of 75 ns (75 × 10^-9 seconds).
  
• The number of turns per meter in the solenoid is to be determined.
  

Step 2: Formula for Magnetic Field in the Solenoid

The magnetic field inside a long solenoid is given by the formula:
B = μ₀ n I

where:
- B is the magnetic field strength,
- μ₀ is the permeability of free space (4π × 10^-7 T·m/A),
- n is the number of turns per meter,
- I is the current in the solenoid.
Substituting the given value of I = 1.5 A, we have:
B = μ₀ n × 1.5

Step 3: Force on the Electron in Circular Motion

The force on an electron moving in a magnetic field is given by the Lorentz force formula:
F = e v B

where:
- e is the charge of the electron (1.6 × 10^-19 C),
- v is the velocity of the electron,
- B is the magnetic field inside the solenoid.
For an electron in uniform circular motion, the magnetic force provides the centripetal force:
F = m v² / r where:
- m is the mass of the electron (9.11 × 10^-31 kg),
- v is the velocity of the electron,
- r is the radius of the circular motion.
Equating the two forces, we get:
e v B = m v² / r
Simplifying, we find the velocity of the electron:
v = e B r / m

Step 4: Relate Time Period to Magnetic Field and Radius

The time period T is the time taken for one complete revolution of the electron, which is related to the velocity and the radius by:
T = 2π r / v
Substituting the expression for v, we get:
T = 2π m / e B Rearranging for B, we have:
B = 2π m / e T Substituting the known values m = 9.11 × 10^-31 kg, e = 1.6 × 10^-19 C, and T = 75 × 10^-9 s, we get:
B = (2π × 9.11 × 10^-31) / (1.6 × 10^-19 × 75 × 10^-9) = 1.58 × 10^-3 T

Step 5: Calculate the Number of Turns per Meter

Now that we know the magnetic field B, we can substitute it into the equation for the magnetic field inside the solenoid to find n:
B = μ₀ n I
Substituting the known values B = 1.58 × 10^-3 T, μ₀ = 4π × 10^-7 T·m/A, and I = 1.5 A, we get:
1.58 × 10^-3 = (4π × 10^-7) n × 1.5
Solving for n, we find:
n = (1.58 × 10^-3) / ((4π × 10^-7) × 1.5) = 250 turns/m

Conclusion

The number of turns per meter in the solenoid is 250.