A proton is shot with a velocity v at an angle  to a uniform magnetic field B, which is directed along the x-axis as shown below. Assume that the only force on the proton is the magnetic force. Explain why the proton will move in a helical path. Find an expression for the pitch of the helix. + B v  x y z Bold type denotes vector quantities

Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11

1. Why does the proton feel a magnetic force? i) Because it has a charge ii) Because it is moving with a velocity which is not parallel to the magnetic field iii) Because it has a mass AA : i only BB : ii only CC : iii only DD : i and ii only E E : ii and iii only

Choice A Incorrect In order to feel a magnetic force, a particle must be charged. There is another condition that must be met for the particle to feel a magnetic force. Think about the following equation: Unit vector perpendicular to v and B

Choice B Incorrect This is true, because if a particle is moving with a velocity that is parallel to the magnetic field (  =0), the particle will not feel a force: BUT, this is not the only reason that the proton will feel a magnetic force. Unit vector perpendicular to v and B

Choice C Incorrect Mass has nothing to do with a particle feeling a magnetic force.

Choice D Correct Notice that if the charge q of a particle or the angle  between the magnetic field and the velocity of the particle is zero, the magnetic force on the particle will be zero. Unit vector in the direction of the force

Choice E Incorrect One reason that the proton feels a magnetic force is because it is moving with a velocity that is not parallel to the magnetic field, but mass has nothing to do with a particle feeling a magnetic force. Try again

We will break the motion of the proton into components to see the contributions of each. Let’s first take a look at the motion of the proton along the x-axis. Notice that the axes are defined on the first slide.

2. Which of the following choices correctly describes the x-component of the proton’s motion? A: A: The proton is accelerating in the x- direction (a x >0) B: B: There is no motion in the x-direction. C: C: The x-component of the proton’s velocity v x is constant.

Choice A Incorrect The proton is not accelerating in the x-direction, because the x-component of the magnetic force F x is zero. The x-component of the proton’s velocity is parallel to the magnetic field, therefore a force will not be felt by the proton due to the horizontal component of motion.  vyvy vxvx v B

Choice B Incorrect Initially, there is motion in the x-direction and the y-direction, because it is stated that the proton is shot with a velocity at an angle to the x-axis. It is not shot directly upwards in the y-direction.

Choice C Correct This is true because there is no x-component of magnetic force on the proton. The x-component of the proton’s velocity is parallel to the magnetic field, therefore a force will not be felt by the proton due to the horizontal component of motion.

3. What type of path will the proton follow if only the motion in the x-y plane is considered? A: A: Straight line (Linear) B: B: Parabolic C: C: Circular This is due to the horizontal component of the initial velocity.

Choice A Correct If we only consider the motion in the x-y plane, the particle will move with constant velocity in the positive x-direction and will not change directions. This describes a linear path.

Choice B Incorrect If we only consider the motion in the x-y plane, the particle will move with constant velocity in the positive x-direction and will not change directions. This describes a linear path.

Choice C Incorrect If we only consider the motion in the x-y plane, the particle will move with constant velocity in the positive x-direction and will not change directions. This describes a linear path.

4. Which of the following choices correctly describes the y-component of the proton’s motion as it moves through the magnetic field? A: A: The proton is accelerating in the y- direction (a y ≠0) B: B: There is no motion in the y-direction. C: C: The y-component of the proton’s velocity v y is constant.

Choice A Correct The proton would accelerate in the y-direction because the y-component of velocity causes a magnetic force to be felt by the proton. Where:  ≠0

Choice B Incorrect There is motion in the x-direction and the y-direction because it is stated that the proton is shot with a velocity at an angle  to the x-axis. It is not shot horizontally.

Choice C Incorrect The y-component of the proton’s velocity v y is not constant, because this motion causes a magnetic force to be felt by the proton. Where:  ≠0

5. What type of path will the proton follow if only the motion of the proton in the y-z plane is considered? A: A: Straight line (Linear) B: B: Parabolic C: C: Circular This is due to the vertical component of the initial velocity.

Choice A Incorrect Using the right hand rule we see that the magnetic force on the proton will cause it to travel on a circular path (counterclockwise if looking down the x-axis from the origin). F B v The force must be centripetal and have constant magnitude to obtain uniform circular motion. At any instant during the proton’s motion, your middle finger should point down the positive x-axis. and your pointer finger should point in the direction of the y- component of the velocity. This will direct your thumb in the direction of the force that pushes the proton.

Choice B Incorrect Using the right hand rule we see that the magnetic force on the proton will cause it to travel on a circular path (counterclockwise if looking down the x-axis from the origin). F B v At any instant during the proton’s motion, your middle finger should point down the positive x-axis. and your pointer finger should point in the direction of the y- component of the velocity. This will direct your thumb in the direction of the force that pushes the proton. The force must be centripetal and have constant magnitude to obtain uniform circular motion.

Choice C Correct If we only consider the motion of the proton in the y-z plane due to its initial vertical motion, the proton’s path would be circular.

6. What type of path do you get if you combine the contributions of each component of the proton’s motion ? A: A: Linear B: B: Helical C: C: Circular

Choice A Incorrect This is the result if only the component of the motion along the magnetic field (x- component) is considered.

Choice B Correct If you trace out a circle while moving down an axis, the resulting path is helical. In this case, the proton’s motion along the field is at a constant velocity

Choice C Incorrect This is only the effect of the proton’s motion in the y-z plane.

7. To find the radius of the projected motion of the proton in the y-z plane, choose all of the statements that are relevant. (i) the magnetic force provides the centripetal acceleration (ii) the magnetic force should be equated to the mass of the proton times centripetal acceleration (iii) the centripetal acceleration is: where R is the radius of the circle A: A: i & ii only B: B: i & iii only C: C: i, ii, and iii

Choice A Incorrect This equation is the correct expression for centripetal acceleration and will be used to find the radius of the proton’s path.

Choice B Incorrect As usual, we set F=ma. In this case the mass is the mass of a proton, and a is centripetal acceleration expressed as:

Choice C Incorrect All of these statements are relevant to finding the radius of the proton’s path.

8. Which of the following expressions is correct for the magnitude of the component of the velocity perpendicular to the magnetic field? A: B: C:

Choice A Incorrect Check your algebra

Choice B Incorrect Check your algebra

Choice C Correct

9. Which of the following expressions is correct for the period T? A: B: C:

Choice A Correct Period:

Choice B Incorrect Period:

Choice C Incorrect Period:

10. Which one of the following expressions gives the pitch P of the helix (the distance traveled in the x-direction during one revolution) ? A: B: C:

Choice A Incorrect The pitch of a helix is the distance traveled along the helical axis during one revolution. The helical axis is the x- axis in this case. We are not concerned with the y-component of motion here.

Choice B Incorrect The pitch of a helix is given by the following expression:

Choice C Correct This is the correct expression for the pitch of the helix since the axis of the helix is the x-axis.

11. Find the expression for the pitch of the helical path of the proton described in the original problem. Answer

Since the x-component of the proton’s velocity is constant: From question 9 we have: