A beam of protons is accelerated from rest inside a straight horizontal tube of length 3.00 m, as shown by region 1 in Figure 1. The acceleration is produced by a potential difference applied between the two ends of the tube. You may assume that this potential difference produces an electric field that is uniform within the tube, and zero beyond the ends of the tube. The beam exits the tube travelling in a horizontal west-east direction, each proton having a kinetic energy of 1.60 x 10^-16 J.
(a) Give a simple argument explaining the direction of the electric field in the tube and explaining which end of the tube is at the higher potential. Having passed well beyond the influence of the tube, the proton beam enters a region of uniform magnetic field (region 2 in Figure 1), which bends it through a circular arc of radius 5.00 cm. You may ignore the Earth's magnetic field. On leaving region 2 each proton is travelling vertically upwards.
(b) State what the direction of the magnetic field must be in region 2, and name the rule you have used to determine it.
(c) Assuming that the proton beam travels in a vacuum and that the effects of gravity can be neglected, what is the kinetic energy of each proton as it emerges from region 2? Explain your answer.
(d) Find the magnitude of the electric field in region 1.
(e) Starting from the formula for magnetic force field the magnitude of the magnetic field in region 2.
(f) Suppose it was necessary to produce a beam of protons with each proton having a final kinetic energy of 3.20 x 10-16 J without altering the geometry of the two regions or the path of the protons. Would it be necessary to change both the electric and magnetic fields?
What electric and magnetic fields would you use?
The links of the figure 1:
- Prof. PhysicsLv 71 decade agoFavorite Answer
a. The direction of electric field in the tube is from west to east through the tube. It is because proton accelerates in the direction the same to the direction of the E-field, and it gains kinetic energy.
The west end of the tube is at a higher potential.
b. The direction of the magnetic field is inside the paper (negative z-axis). The rule is Fleming's left hand rule.
c. The kinetic energy is unchanged, it is 1.6 X 10-16 J. It is because the magnetic field does not do work on the proton.
d. By eV = K.E. gained
(1.6 X 10-19)V = 1.6 X 10-16
Potential difference, V = 1000 V
Since the electric field is uniform,
Electric field, E = V/d = 1000/3.00 = 333 NC-1
e. K.E. = 1/2 mv2
1.6 X 10-16 = 1/2 (1.67 X 10-27)v2
Speed, v = 4.38 X 105 ms-1
By Lorentz force,
qvB = mv2/r
Magnetic field, B = mv/qr
= (1.67 X 10-27)(4.38 X 105)/(1.6 X 10-19)(0.05)
= 0.091 T
f. It would be necessary to change both the electric and magnetic fields.
The electric field used is directly proportional to the speed of the proton. So, the electric field should be doubled, it is 667 NC-1.
And for the magnetic field, it is also directly proportional to the speed of the proton. Without changing the geometry (radius), the magnetic field used should also be doubled, it is 0.183 TSource(s): Physics king