A singly charged ion of unknown mass moves in a circle of radius 12.5 cm in a magnetic field of

A singly charged ion of unknown mass moves in a circle of...

Key Concepts

Lorentz Force

This is the force experienced by a charged particle moving in a magnetic field, acting perpendicular to both the velocity of the particle and the magnetic field. It is given by the vector equation F = q(v × B) and is the key factor that causes a charged particle to undergo circular motion when its velocity is perpendicular to the magnetic field.

Centripetal Force

In uniform circular motion, a net inward force, known as the centripetal force, must act on a particle to maintain its curved path. For a mass m moving with speed v in a circle of radius r, the centripetal force is given by F = m*v^2/r. When a charged particle moves in a magnetic field, the Lorentz force provides this centripetal force.

Kinetic Energy from Electric Potential

When a charged particle is accelerated through an electric potential difference, its electric potential energy is converted into kinetic energy. The energy conversion is expressed by the relation 1/2*m*v^2 = q*V, where q is the charge of the particle and V is the potential difference. This relation is used to determine the speed of the particle after acceleration.

Combining Energy and Force Equations

By integrating the concepts of energy conversion and centripetal motion, one can eliminate the velocity parameter to relate the mass of the particle directly with measurable quantities such as the radius of the circular path, the magnetic field strength, and the potential difference. This method involves substituting the expression for velocity from the energy conversion equation into the equation for the radius of circular motion.

Transcript

00:01 In this question, we are told that there is a singly charged ion of unknown mass.

00:07 So when an ion is singly charged, that just means it will have a charge of magnitude 1 .6 times 10 to the negative 19 coulum.

00:17 Because singly charged can either mean you have one more electron than you're supposed to or one fewer electron than you're supposed to.

00:26 So either way, the charge differential will be 1 .6 times 10 to negative 19th coulums.

00:34 And it tells us that it is moving in a circular path of radius 12 .5 centimeters, which can be converted to 0 .1 to 5 meters.

00:46 And it is moving through a field of 1 .2 tesla's.

00:52 And before it enters this field, this ion was accelerated through a potential difference.

01:00 So it's kind of like an velocity accelerator.

01:04 So the, and i will try to make the v for electric potential very different from velocity.

01:14 Okay, so delta v standing for potential difference.

01:17 In this case, it's 7 .0 kilovolts.

01:22 Converting that to volts, you have 7 ,000 volts.

01:27 And because they said at the beginning, this ion had an unknown mass, it shouldn't be a surprise to us that they're looking for the mass of this ion.

01:36 So before we can solve anything here, we must use what we know about electric potential to solve for, velocity first.

01:52 So when i talk about velocity, i'm talking about the velocity that the ion had when it first entered the magnetic field.

02:02 So in other words, if this is how, if this is the region of potential difference, right? so by the time the ion enters from this point and by the time it leaves this point to a uniform magnetic field, what is the velocity at that point? okay.

02:22 So this deals with the conservation of energy because when it was when it first entered this region of acceleration of potential difference, all it had was electric potential energy.

02:39 And by the time it left this region of potential difference, all i had was kinetic energy.

02:45 Or another way you can think of it is whatever potential energy, electric potential energy was lost.

02:50 As they went from one end to the other end across this potential difference was gained, was how much kinetic energy they've gained by the time they leave this region.

03:04 So either way you look at it, it's fine.

03:07 It hinges upon the concept of conservation of energy.

03:10 But how is electric potential energy related to electric potential? well, hopefully you remember from previous units that potential energy is related to electric potential by u is equal to q times v.

03:27 Because the very definition of electric potential is the potential energy per unit charge, right? so rearranging that, you have u equals qv.

03:38 And so i'm going to write that here.

03:40 Again, please don't confuse this with velocity.

03:43 This is a capitalized v.

03:45 And your kinetic energy, as always, is equal to 1 .5m v squared.

03:51 I notice there's a little tail right here, so this is little v standing for velocity.

03:56 And again, my goal of doing this is to solve for velocity.

04:01 So if i rearrange everything, i will find that velocity is equal to square root of 2 q times essential difference divided by mass.

04:12 Now, are we ready to solve for what velocity? is before moving on with the next part of this question? well, the answer is no, because we don't have mass.

04:22 And mass is what we're ultimately looking for.

04:24 So it looks like we have to leave velocity as an expression for now.

04:29 Okay, so how do we, where do we go from here, right? we found the velocity, but our goal is to look for the mass.

04:38 So that's when you have to recognize that by the time this ion leaves this potential difference region, it will enter another region of uniform magnetic field.

04:49 And because there's only magnetic field there, that means it will only experience a magnetic force, which does no work on the ion, and it only changes its direction.

05:00 So i am just arbitrarily drawing a circular path here.

05:05 What i'm trying to say is it will experience a radial or centripetal acceleration from the centripetal force, which is the magnetic force in the situation.

05:16 So you can set equality between centripetal force formula and the magnetic force formula.

05:22 And that's exactly what we're going to do next...

Please give Ace some feedback