A horizontal beam of alpha particles are injected with a speed of 1.3x10^5 m/s into a region with a vertical magnetic field of magnitude 0.155T.
A- How long does it take for an alpha particle to move half way through a complete circle?
B- If the speed of the alpha particle is doubled, does the time found in part A increase, decrease, or stay the same? Explain.
C- Repeat for part A for alpha particle with a speed of 2.6x10^5 m/s.
D- What does this infer with regard to the frequency of EMR radiated by identical charges (at any speed) entering a perpendicular magnetic field?
Malus's Law
I=Io•(cosφ)^2,
cos φ =sqrt(I/Io),
φ =19.19o
I'm sorry. I've mistaken. This is solution of another problem
A. Lorentz force
F=q•v•B•sinα,
sin α =1.
mv^2/R = q•v•B,
R= mv/qB.
The period is
T =2•π•R/v = 2•π•m/q•B.
For α-particle
T = 2•π•4m(p)/2•e•B.
T/2 = 2•π• m(p)/ e•B,
where m(p) = 1.66•10^-27 – the mass of proton
e = 1.6•10^-19 C is the charge of proton.
B. T doesn't depend on velocity.
F=q•v•B•siná,
sin á =1.
mv^2/R = q•v•B,
R= mv/qB.
The period is
T =2•ð•R/v = 2•ð•m/q•B.
For á-particle
T = 2•ð•4m(p)/2•e•B.
T/2 = 2•ð• m(p)/ e•B
A- To calculate how long it takes for an alpha particle to move halfway through a complete circle, we need to find the time period (T) taken for one complete revolution and then divide it by 2.
The time period of a charged particle moving in a magnetic field can be found using the formula:
T = (2πm) / (eB)
Where:
- T is the time period,
- π is pi (approximately 3.14),
- m is the mass of the particle,
- e is the charge of the particle, and
- B is the magnitude of the magnetic field.
For an alpha particle, the charge (e) is 2 times the elementary charge (1.6x10^-19 C) and the mass (m) is 4 times the mass of a proton (4x1.67x10^-27 kg).
Substituting the values given into the formula, we can calculate the time period (T). Finally, to find the time taken to move halfway, divide T by 2.
B- If the speed of the alpha particle is doubled, the time period and the time taken to move halfway through a complete circle will stay the same. This is because the formula for the time period does not depend on the speed of the particle; it only depends on the charge, mass, and magnetic field strength. Increasing the speed doesn't affect the magnetic force acting on the particle, thus not affecting the time it takes to complete a circle.
C- To calculate the time taken for an alpha particle with a speed of 2.6x10^5 m/s to move halfway through a complete circle, we follow the same steps described in part A. We use the new speed value in the calculation instead of the initial speed provided. Thus, the calculation will give us the updated time taken.
D- The frequency of electromagnetic radiation (EMR) radiated by charged particles entering a perpendicular magnetic field is directly proportional to the speed of the charges. This implies that as the speed of the charges increases, the frequency of the EMR also increases. The classical theory of electromagnetism predicts that accelerating charged particles emit electromagnetic waves. Therefore, higher speeds of the charges result in a higher frequency of EMR radiated.