Given Data
- Turns per meter of solenoid: 2 * 10^4 turns/m
- Diameter of the solenoid: 10 cm (radius = 5 cm = 0.05 m)
- Kinetic Energy (K.E.) of electron beam: 100 keV
Understanding the Problem
To find the current in the solenoid that allows the electron beam to pass through without touching the walls, we first need to calculate the magnetic field inside the solenoid and then relate it to the velocity of the electrons.
Calculating Magnetic Field
- The magnetic field (B) inside a solenoid is given by the formula:
B = μ₀ * n * I
where:
- μ₀ = 4π * 10^-7 T*m/A (permeability of free space)
- n = number of turns per unit length (2 * 10^4 turns/m)
- I = current in the solenoid (A)
Finding Velocity of Electrons
- Kinetic Energy (K.E.) is given by:
K.E. = 1/2 * m * v²
- Rearranging, we find:
v = sqrt(2 * K.E. / m)
where m (mass of an electron) = 9.11 * 10^-31 kg and K.E. = 100 * 10^3 eV (1 eV = 1.6 * 10^-19 J).
Setting Up the Equation
- The magnetic force acting on the electrons provides the centripetal force required for circular motion:
F_magnetic = e * v * B = m * v² / r
- Here, e (charge of an electron) = 1.6 * 10^-19 C, and r (radius of the solenoid) = 0.05 m.
Finding the Current
- By combining the equations and solving for I, we can find the required current in the solenoid. After substituting values, we can compute I numerically.
In conclusion, by following these calculations, we can derive the current needed in the solenoid to ensure the electron beam passes through without touching the walls.