When a charged particle moves at an angle of 25^∘ with respect to a magnetic field, it experienc

When a charged particle moves at an angle of 25^∘ with...

When a charged particle moves at an angle of $25^{\circ}$ with respect to a magnetic field, it experiences a magnetic force of magnitude $F$ At what angle (less than $90^{\circ}$ ) with respect to this field will this particle, moving at the same speed, experience a magnetic force of magnitude 2$F ?$

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field. It is given by the cross product of the velocity of the particle and the magnetic field, indicating that the force depends on the sine of the angle between these two vectors. This dependence explains why the magnitude of the force changes with the angle of motion.

Sine Function in Cross-Product

In the expression for the Lorentz force, the sine function quantifies how the force varies with the angle between the velocity vector and the magnetic field direction. Since the sine of an angle determines the magnitude of the component of one vector perpendicular to another, adjusting the angle appropriately allows control over the magnitude of the force acting on the particle.

Transcript

00:01 Hello.

00:02 In this problem, we are asked to consider two different situations.

00:07 One, we have a particle moving at a 25 degree angle at some velocity v with respect to a magnetic field, b.

00:24 Now, this angle we're given is 25 degrees.

00:29 Now, we know that the magnetic force that this particular particle feels while traveling at this particular angle is equal to f.

00:43 Now, we're asked to consider another situation, one in which the angle has changed, but everything else, mostly everything else, has stayed the same.

00:53 So the object is going to be traveling at a different angle, but the magnetic field is still say this way, with the same magnitude, the particle is traveling at the same magnitude of velocity, but now we're traveling at an unknown angle.

01:09 We do not know what this angle is.

01:12 But we do know that the magnitude of the magnetic force that the particle feels in this scenario, which i'll remember to draw my little particle here, both of charge, let's say, q0, we know that this particle in this lower situation is going to be feeling a force of.

01:31 Twice the amount as before.

01:34 So the question is, what is this new angle that the particle is traveling at? well, what we're going to do is analyze both of these situations individually and then link them through some variables that we're going to solve for.

01:48 So for our top situation, let me kind of draw a line between the two.

01:52 Let's write out what the magnitude of the magnetic force would be in terms of variables.

01:57 So we know that the magnitude of the magnetic force is given by the magnitude, of the charged particle, magnitude of the charge of the charged particle, multiplied by its speed, then by the magnitude of the field through which the particle is traveling, and then multiplied by sign of the angle between the magnetic field and the velocity vectors.

02:19 Now, this we know gives the magnitude of the magnetic field, and that is represented by our letter capital f here.

02:27 So this is the statement for the magnitude of the magnetic force for our particle in the first scenario.

02:36 Now, let's write out the magnitude of the force the particle is going to feel in this second scenario.

02:41 We still have the same charge here...

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