(a) An ^16 O nucleus (charge +8 e ) moving horizontally from west to east with a speed of 500 km

(a) An ^16 O nucleus (charge +8 e ) moving horizontally from...

Key Concepts

Minimum Conditions and Trade-offs in Parameters

The concept of minimum conditions, such as the minimum velocity required to produce a given magnetic force, is based on maximizing the sinusoidal factor (i.e., setting the angle to 90°). However, alternative combinations are possible; increasing one parameter (such as the magnetic field strength) can compensate for a less-than-optimal condition in another (like a reduced velocity or an angle that is not exactly 90°), while still producing the same magnitude and direction of force.

Dependence on the Angle Between Velocity and Magnetic Field

The magnitude of the magnetic force is directly proportional to sin(?), where ? is the angle between the velocity and the magnetic field. When the particle’s motion is perpendicular to the magnetic field (? = 90°), the force is maximized. For any other angle, the force is reduced, indicating that adjustments in the field strength or velocity direction can achieve the same force magnitude under different conditions.

Lorentz Force

The Lorentz force is the fundamental principle that describes the force experienced by a moving charged particle in the presence of a magnetic field. This force is given by the equation F = q(v × B), meaning the magnitude of the force depends on the charge, its speed, the strength of the magnetic field, and the sine of the angle between the velocity and the magnetic field vectors.

Vector Cross Product and Right-Hand Rule

The cross product in the Lorentz force formula indicates that the force is always perpendicular to both the velocity of the particle and the magnetic field. The right?hand rule is used as a tool to determine the direction of this force, by aligning the fingers in the direction of the particle’s velocity and curling them toward the magnetic field, with the thumb pointing in the direction of the force.

Transcript

00:01 You know that force, magnetic force, is given by q times the cross product of velocity and magnetic field.

00:12 So this means that the magnetic field is perpendicular to force.

00:22 And if force is non -zero, the magnetic field cannot be parallel to the velocity.

00:31 So basically the component of magnetic field contributing to the force should be perpendicular and not parallel to the velocity vector.

00:41 Now let's apply it to the electron and proton.

00:48 So for the proton, so this is the first part.

00:54 So let's say that b is the magnetic field has all three components.

01:04 Now for the proton we see that force is along the y direction and velocity is along x direction.

01:20 This means that the magnetic field should be perpendicular to the force or j -cap and magnetic field is not parallel to the velocity.

01:36 I cap so basically the magnetic field contributing to the force so not the total magnetic field this means that so if this is if the component of magnetic field contributing to force there's perpendicular to j cap this means that we have two choice i cap and k cap but at the same time it should not be parallel to i cap so we just have one choice which is k cap this means that bz is the one taking part in the magnetic force for the proton.

02:27 Now let's move on to apply that.

02:31 So fp, let's call it fpj cap, where fp is just the magnitude of the force, is equal to q times v.

02:42 So this is basically vp, which is vxi.

02:46 Gap because there is no component alum y and z direction and the magnetic field which in this case is bz k gap because all other components basically go to either zero or give the force in some other direction than y gap so this will give q times vx times v z z just a minute bc right oh sorry bc times i cap cross k cap this is basically negative j gap so negative q v x bc bc sorry j gap and from here we can see that the magnitude is equal to negative q v x bc bc this gives the magnitude of magnetic field along c direction to be fp over negative fp over q times vx we solve it and we get this value to be one point this is negative 0 .9375 tesla.

04:38 This means that the magnetic field has a complete.

04:44 Of has a component along the negative z direction because of the negative sign and has a magnitude of this along that direction.

04:59 Now it will be clear after we find after we do the same thing for the electron.

05:09 So for the electron, the force is along y -cap and velocity is.

05:18 Is a k -cap.

05:19 This means that for the electron magnetic field should be perpendicular to j -cap and not parallel to the velocity which is along k -cap.

05:48 Let's not take the negative sign here into account because it doesn't affect much.

05:56 We are anyway going to take in this while calculating the magnetic field.

06:05 So here we see that there can only be two options for this icap and k -cap, but it cannot be parallel to k -cap.

06:14 That means b -cap is a long i -cap.

06:22 So let's apply it now.

06:24 So ve is equal to q times ve vector which is negative ve.

06:34 X k cap so let's write it that way negative v x k cap cross product with vx because there is no component so b y and b z do not contribute in this force and this is basically v e j cap which i forgot to write so this is v x i cap which gives q times negative q times vx xxx times so this is z minus tech v x k cross i cap and k cross i is j caps.

07:25 So this is negative q times v z times v x j cap.

07:34 This gives bx to b f e over negative v over q times v z times v z and we get this value...

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