A particle has a charge of q=+5.60 μC and is located at the coordinate origin. As the drawing sh

A particle has a charge of q=+5.60 μC and is located at the...

Key Concepts

Role of Velocity in Magnetic Interactions

Velocity is a critical factor in determining the magnetic force acting on a charged particle. Since the magnetic component of the Lorentz force depends on the cross product of velocity and the magnetic field, both the speed and the direction of the particle's motion significantly influence the magnitude and direction of the resulting force. This concept is key when analyzing scenarios where the particle changes its direction of motion relative to the magnetic field.

Lorentz Force

This is the total force acting on a charged particle moving in both electric and magnetic fields, and is given by F = q(E + v × B). It combines the effects of both electric and magnetic forces into a single framework, and its evaluation requires careful vector addition of the components arising from each field.

Vector Cross Product

The vector cross product is an operation used to determine a vector that is perpendicular to two given vectors. In the context of electromagnetic forces, it is used to calculate the magnetic force via the term v × B. The magnitude and direction of the resulting vector depend on the magnitudes of the original vectors and the sine of the angle between them, which is fundamental to determining the correct direction of the magnetic force.

Electric Force

This is the force experienced by a charged particle in the presence of an electric field. It is defined by the relation F = qE, where q is the charge and E is the electric field vector. The direction of this force is determined by the sign of the charge relative to the direction of the electric field, and it acts regardless of whether the charge is moving or stationary.

Magnetic Force

The magnetic force acts on a moving charged particle when it is in the presence of a magnetic field. It depends on the particle's velocity, the magnetic field, and the charge, and is given by F = q(v × B). This force is perpendicular to both the velocity of the charge and the magnetic field, and it only manifests when the charged particle is in motion.

Transcript

00:01 Alright, so for this problem, the setup is, the situation is that we have a particle, a charged particle, and it's centered at the origin for a three -dimensional axis.

00:17 So we have our z, we have our y axis, and we have our x axis.

00:46 And we have, so this is a three -dimensional axis here, negative z going down, negative x going to the left, and negative y coming out of the page.

01:03 So positive y goes into the page here.

01:08 And we have three fields.

01:10 We have one electric field, e, and that's in the positive x direction.

01:22 We have a magnetic field also in the positive.

01:25 Of x direction.

01:32 I'm going to call that one, unsurprisingly, bx.

01:38 That's the magnetic field in the x direction.

01:40 We also have a magnetic field in the positive y direction, so into the page.

01:46 I'm going to call that by.

01:53 So we have an unmoving particle here at the center at the origin, and we have these three fields.

02:02 And we want to find, in part a, we want to find the magnitude and direction of the force on the particle.

02:08 So we have two different types of forces here, or we have two different types of fields causing forces, so we'll need the equations for both of those.

02:18 Force for electric field is an easy equation.

02:21 The force is equal to the charge times the electric field.

02:34 That's another thing we're given here is the charge q.

02:37 We know the charge.

02:38 It is a positive charge.

02:45 So this is easy to calculate.

02:47 For part a, we have no speed.

02:53 I'm sorry, a.

02:55 For part a, we have v equals zero.

03:00 No velocity.

03:05 Oh, we need to have our other equation here.

03:08 So we have magnetic field force is equal to charge times the velocity, times the magnetic field, sine of theta, sine of the angle between the traveling velocity and the magnetic field.

03:33 Those are the two vectors in this equation.

03:37 Velocity and magnetic field.

03:39 Those are the universal sign for a vector.

03:43 Q is a scalar quantity.

03:47 So now back to part a, we have no velocity.

03:49 The particle is stationary here.

03:52 So for the two magnetic fields, by and bx, we have no velocity, which means this v here is zero, which means the force is zero.

04:00 We can't have a force from a magnetic field unless we have a moving charge.

04:05 So we have no forces from by or bx.

04:08 E, however, does not have the same stipulation.

04:10 There's no v in this equation.

04:13 So we can find f .e, the force from the electric field, is simply the charge times the electric field.

04:25 Qe.

04:27 Now what about direction? the force, an electric field, are directed in the same direction, positive x -axis.

04:40 So we have our direction here is positive x, which relevant to this problem, the positive x -axis is to the right.

04:51 This grid here shows that you can define an axis any way you want.

04:55 We could have the x -axis flowing out of the page.

04:58 Oh, we could have the negative x -axis pointing down.

05:00 It doesn't matter.

05:01 It all depends on how you define it, and then you have to stick to it.

05:06 So we have positive is to the right for x, which means everything we do in that direction has to be positive.

05:11 And same for positive y.

05:13 It's backwards into the page, which means everything in that direction has to be positive, as far as this problem goes.

05:22 So that's part a.

05:24 The only force on it is from the electric field, charge times the magnitude of the electric field, and the force is in the same direction as the electric field, which is in the plus x direction.

05:35 So that's part a.

05:43 For part b, we have a velocity now that we're given, v, and we're told v is in the plus x direction, which means moving to the right.

06:02 So let's take a look here and see which one of these, see how many of these fields will affect it.

06:10 So for e, e is independent on any, factor of sign so e isn't going to be eliminated we'll have the same number here our velocity is in the same direction as e whereas the the uh...

06:27 Or our our field for e is in the same direction as the velocity which means because we have a sign of theta here that this bx is parallel to v when we have a parallel velocity to our magnetic field parallel means that theta here is zero or if it's anti -parallel, it would be 180.

06:47 But in either case, sign of 0 and sign of 180 is 0.

06:52 So there's no force from this bx quantity.

06:55 So the forces that we have, the total force is equal to, i'll call it ft, is equal to by.

07:07 We have a force from the y, because by the right -hand rule, we have our thumb pointing in v, which is long, the positive x -axis are fingers flat and pointing in the direction y which means we'll have a force up from this b y b -y will be up and we'll have a force e pointing to the right so we'll have two forces b x will be zero because for the case of the y why traveling into the page and the velocity traveling along the positive x -axis and make a right angle.

07:51 So they are perpendicular which means we have sign of or we have sign of 90 degrees which is one.

07:56 So that one will give us a force.

08:01 So our total force is equal to by plus e.

08:12 We could call it ex just to keep with our sub subtext here.

08:18 The electric field in the extraction which is the only electric field.

08:21 That's just simply the force here.

08:23 And that will be the same because we have the same charge, and we have the same electric field.

08:29 Velocity doesn't play a factor into the force caused by the electric field.

08:33 If the velocity were changing, then we would have an acceleration and we'd have an outside force other than what we have here.

08:39 But it's a constant velocity, which means there's no outside force.

08:43 So that doesn't affect our force here.

08:46 So this is the same thing we calculated in part a, this fe.

09:00 So we know that.

09:01 One already now we need to find this by so what does by by is the force from the magnetic field in the y direction so this is fy and that's what we need to find using this equation we have a force due to the magnetic field in the y direction is equal to charge times the velocity times the magnetic field and we have all of those so this is something we can calculate.

09:44 This is b -y.

09:46 Only the magnetic field in the y direction...

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