Free Charge Moving in Uniform Magnetic Field

[prslook26]

Homework Statement

The drawing shows a particle carrying a positive charge +q at the origin (of x and y axis), as well as a target location located in the lower left quadrant. The target is just as far from the x-axis as it is from the y axis. There is also a uniform magnetic field directed away from you. Our goal is to start the charge moving in the correct direction, and with the correct speed, so that the magnetic force on the charge will cause it to hit the target.

(a) In which direction(s) could the charge begin moving and, assuming the correct speed, reach the target location? You have four choices for the direction: +x, -x, +y, or -y. Note that more than one answer may be correct. Explain your reasoning.

(b) Assign any values you like for the magnetic field strength B (in Tesla), the charge q (in Coulombs), and the coordinates of the target point (in meters).

What must the speed of the charge be in order to reach the target location? Give your answer in m/s. (Note that your answer will be the same no matter which answer you gave in part (a) above, provided you answered part (a) correctly.)

Homework Equations

The Attempt at a Solution

(a) I apologize for not being able to provide a graph for this problem, I couldn't find it anywhere online.

So, since the charge is positively charged, it will start moving counterclockwise in the uniform magnetic field, which will be in the negative direction for both x and y-axis - in other words towards the target, which is in the lower left quadrant. But since I wasn't given exact coordinates of the target, can I simply assign my own numbers (say: -3,-3) for the target and say that the charge needs to move -3 on the x-axis and -3 on the y-axis in order to reach the target?
I feel like this sounds too simple to be correct!

(b) In order to find the speed of the charge, we assume that the magnetic force equals its centripetal force:

Fm = Fc
qV1B = m x V1^2/r

V1 = rqB/m

If target is at (-3,-3), then it's really a square, so the radius is simply:

a^2+b^2=c^2
c=4.24 m (square root of 18)

Let's also assume:
+q = 1.60x10^-19 C
m = 1.67x10^-27 kg
B = 4x10^-4 T

Solving for speed:

V = [(4.24m)(1.60x10^-19 C)(4x10^-4 T)] / 1.67x10^-27 kg
V = 1.62 x 10^5 m/s


Am I even close with this? I have been staring at this problem for hours and this is all that I'm coming up with. Help!

Thank you in advance!

 

[haruspex]

can I simply assign my own numbers (say: -3,-3) for the target and say that the charge needs to move -3 on the x-axis and -3 on the y-axis in order to reach the target?

Those are the displacements that are required, but what direction does it have to move off into get there? Suppose you just send it off in some arbitrary direction with some arbitrary speed. What will its subsequent path look like, geometrically speaking?

 

[prslook26]

Thank you so much! I think I've got it now:

The path of the charged particle will be shaped by the Lorentz Force F=qVB, which will act perpendicular to the particle's velocity. Geometrically speaking, the particle will move in a uniform counterclockwise circular motion perpendicular to the magnetic field. And the direction of the Force can be found using the right-hand rule applied to the perpendicular component of the velocity. So in order to reach the target in the lower left quadrant, the charged particle must be sent in either +y or -x direction. Am I right? :)

Also, what about part (b)? Am I on the right track with that?

Again, thank you so much!

 

[haruspex]

The path of the charged particle will be shaped by the Lorentz Force F=qVB, which will act perpendicular to the particle's velocity. Geometrically speaking, the particle will move in a uniform counterclockwise circular motion perpendicular to the magnetic field.

I can never remember which way things are defined, so I don't know whether it would be clockwise or counterclockwise, but yes, it would be a circle.

So in order to reach the target in the lower left quadrant, the charged particle must be sent in either +y or -x direction.

If I'm reading the question correctly, they want a complete list of possible quadrant directions. The first thing is to figure out what determines the radius of the circle. Is there only one possible radius, or can you vary it by something within your control? (You can set the direction and the speed.) Having decided that, consider all the possibilities for that circle, given the points you know it must pass through.
Also, I don't understand how you picked +y and -x. Isn't the target at (-c, -c) for some c?

Also, what about part (b)? Am I on the right track with that?

If target is at (-3,-3), then it's really a square, so the radius is simply:
a^2+b^2=c^2
c=4.24 m (square root of 18)

The origin and target do not necessarily form a diameter of the circle.

 

[prslook26]

If I'm reading the question correctly, they want a complete list of possible quadrant directions. The first thing is to figure out what determines the radius of the circle. Is there only one possible radius, or can you vary it by something within your control? (You can set the direction and the speed.) Having decided that, consider all the possibilities for that circle, given the points you know it must pass through.
Also, I don't understand how you picked +y and -x. Isn't the target at (-c, -c) for some c?

As far as I understood from his instructions, he wants to know in which direction can we flick the charge so that it will reach the target at (-c,-c). I don't believe he wants a complete list of possible quadrant directions - only whether the charge should be sent off in +x, -x, +y, or -y direction.

As for how I picked +y and -x...I picked them based on the direction a positively charged particle normally takes in a uniform magnetic field: circular and counterclockwise. So, if we sent it up along the y-axis (+y), the perpendicular magnetic force would steer it to the left along the x-axis and as it moved that way, along its circular trajectory it would hit the target at (-c,-c). The same thing if we sent it to the left, in the -x direction; the perpendicular magnetic force would steer it down, along the y-axis, and as it kept spiraling down on its circular trajectory, it would eventually hit the target.

The origin and target do not necessarily form a diameter of the circle.

Is one of the coordinates (he said they're equal) the radius of the circle? Because he didn't say anything specific about the radius - only that we assign any value for the coordinates of the target point - it makes me think that coordinates have something to do with it?

I apologize for all these questions. Physics is not my forte. Thanks again, you've helped me so much already!

 

[haruspex]

As far as I understood from his instructions, he wants to know in which direction can we flick the charge so that it will reach the target at (-c,-c). I don't believe he wants a complete list of possible quadrant directions - only whether the charge should be sent off in +x, -x, +y, or -y direction.

The OP says:

(a) In which direction(s) could the charge begin moving and, assuming the correct speed, reach the target location? ... Note that more than one answer may be correct.

So I read that as wanting the complete list. (And it makes sense to me that that would be the question being asked.)

Is one of the coordinates (he said they're equal) the radius of the circle?

It'll be a circle that passes through the origin (obviously) and the target, (-c, -c). So the straight line from O to target forms a chord of the circle.
In the OP you had an equation relating the radius of the circle to the field, the charge, and the speed. What was it? Which of those parameters are you free to vary? What range does that allow for the radius?