Physics Magnetic Field

Question

An oxygen-16 ion with a mass of 2.66·10−26 kg travels at 5·106 m/s perpendicular to a 1.20

T magnetic field, which makes it move in a circular arc with a 0.231 m radius. What positive

charge is on the ion? (b) What is the ratio of this charge to the charge of an electron?

Answer 1

Roycee. I'm happy to show you how to work through a problem like this.

In this problem we have a particle (an oxygen ion) that is in motion with a known velocity (we know the speed and we can assume a direction - since we're told the oxygen ion is traveling perpendicular to the magnetic field that establishes relative coordinate axes and that's enough for this problem). It encounters a magnetic field and we're told that it begins to travel in a circular arc with known radius. From this we're ask to derive the positive charge on the oxygen ion and compare its magnitude to the charge of an electron.

So here is what we're given: (bold variables (like v) are vectors)

v is 5x10⁶ m/s. (By the way, I think you meant 5x10⁶ m/s and not 5.106 m/s). We are free to choose this direction as long as we maintain v and B as being perpendicular to each other.

B is 1.2 T. It's direction is in either direction along a line perpendicular to v.

m is 2.26x10^-26 kg and is the mass of the oxygen ion.

r is 0.231 m and is the radius of the arc that the oxygen ion travels on once it enters the B field.

q is the (positive) charge of the oxygen ion and is the quantity we want to solve for.

There are two concepts/principles needed to solve this problem:

1) The Lorentz force. The Lorentz force on a charged particle is the force imparted to a charged particle when it is subjected to an electromagnetic field. Mathematically:

F=qE+vxB where x denotes the vector cross product, E is the electric field, B is the magnetic field, and q is the charge of the particle.

In this case, we can assume that E and B are static (non-time-varying) and we can assume that E=0. Weare also told that v is perpendicular to B so vxB is simply the magnitude of v times the magnitude of B with a direction that is perpendicular to both v and B. A handy mnemonic for establishing the direction of a cross product is the "right-hand rule" - if c=axb, the direction of c is given by taking the fingers of your right hand and aligning them in the direction of a. Then curl your fingers in the direction of b. At that point your right thumb will point in the direction of c.

However, for this problem, it's enough to know that v and B are perpendicular. Let's assume however that v is alongThe magnitude of vxB is vBsin(θ) where θ is the angle between v and B which is π/2 or 90°. So in this problem the magnitude of vxB is simply vB. So the Lorentz force on the oxygen ion is:

F=qvB with a direction given by the right-hand rule. If we assume that the direction of v is along the positive x-axis and the direction of B is along the positive y-axis then the right-hand rule tells us that vxB is along the positive z-axis. So the Lorentz force imparted to the oxygen ion is along the positive z axis with a magnitude of qvB.

2) The second concept is from mechanics. In this case we have a particle (the oxygen ion) travelling in a direction (the positive x-axis) with a speed v and a force (the Lorentz force) qvB exerted on it in a direction along the z-axis, which is perpendicular to the x-axis and hence perpendicular to the direction of the motion of the oxygen ion. From mechanics, we know that when a force acts on a particle perpendicular it is centripetal. Hence:

F=mv²/r with a direction perpendicular to v and where m is the mass of the particle and r is the radius of the arc of travel. Since F and v are perpendicular F does no work on the particle and if we assume no drag or friction the particle's kinetic energy remains constant so v remains constant. So the motion of the particle when subjected to a centripetal force is circular.

So pulling the two concepts together we have a Lorentz force on the oxygen ion F=qvB along the z-axis which causes a centripetal motion of the oxygen ion in the B field.

So the Lorentz force is also the centripetal force - this is the crux of the problem:

F=qvB along z-axis; and

F=mv²/r also along z-axis.

We know v,B,m, and r. We want to determine q. Since we know the direction of the F we can just work with scalar quantities. So equating the magnitude of the Lorentz force with the magnitude of the centripetal force gives us:

qvB=mv²/r

Solving for q:

q=mv/Br

We can now solve for q. First we check to make sure the units of all quantities are in the same system of units: m is given in kg, v is given in meters per second, B is given in T (Tesla), and r is given in meters - all are MKSA system units. So we are good to just do the arithmetic.

Solving for q we find:

q=4.80x10^-19 C (coulombs). This answers the first part of the problem.

Next, they ask for the ratio of the oxygen ion mass versus that of the electron (-1.60x10^-19 C) the magnitude of the electron charge is also known as the fundamental charge in nature - also called "e". So really they want to know the oxygen ion charge in terms of the fundamental charge.

So

Q=q/e=3

So the oxygen ion has a fundamental charge of 3e. We can also say that the oxygen ion is triply (3 times) ionized.

I hope this helps!