At a certain point on the surface of the Earth in the southern hemisphere, the Earth's magnetic

At a certain point on the surface of the Earth in the...

Key Concepts

Lorentz Force

This is the force exerted on a charged particle moving in a magnetic field. Its magnitude depends on the particle's charge, the speed at which it is moving, the strength of the magnetic field, and the sine of the angle between the particle’s velocity and the magnetic field. This concept lies at the heart of electromagnetism and explains phenomena ranging from the motion of charged particles in space to the workings of electric motors.

Vector Cross Product

This is a mathematical operation used to determine a vector that is perpendicular to two given vectors. In the context of the magnetic force, the cross product of the velocity vector and the magnetic field vector yields the direction of the force on the particle. The magnitude of this product involves the sine of the angle between the vectors, which plays a crucial role in calculating the force's strength.

Right-Hand Rule

This is a method used to determine the direction of the resultant vector produced by the vector cross product. For charged particles in a magnetic field, the rule helps identify the direction of the magnetic force by aligning the fingers with the first vector (velocity), curling them toward the second vector (magnetic field), and having the thumb point in the direction of the force. This mnemonic is essential for understanding the spatial orientation of forces in electromagnetism.

Charged Particle Motion in Magnetic Fields

This concept encompasses the behavior of particles with electric charge when they move through a magnetic field. The interplay between the particle’s velocity and the magnetic field results in forces that can alter the trajectory, causing the particle to move in curved paths such as circular or helical orbits. Understanding this motion is crucial for applications in particle accelerators, cosmic ray detection, and various technologies involving plasmas and electromagnetic confinement.

Transcript

00:01 In this question, they tell us that at a certain point in the southern hemisphere, you have a magnetic field with the magnitude of 5 .0 times 10 to the negative 5th tesla.

00:15 And that if you were looking at the direction of the magnetic field from the earth's surface, and let's say you have a horizontal surface like this, and you have sort of like a compass, here, then what would happen is that it will show that the magnetic field is pointing north.

00:42 So this is north and that is south.

00:45 And the compass will also have tilted upward above the horizontal at a degree of 55 above the horizontal.

00:57 So basically based on what i've drawn here left is north.

01:03 North and right is south and that not only is there like one plane where you have all four directions, there is also up and down.

01:16 So this is saying that while it is going north, it also is at an angle that is 55 degrees above the horizontal.

01:25 And so likewise, if this is north and that is south and the magnetic field is pointing north, then into the page will then be east and out of the page will be west.

01:38 So let me go ahead and denote that here real quick.

01:41 Into the page will be east and out of the page will be west.

01:48 All right.

01:49 So now that we have a visual for the magnetic field, let's keep working through the other givens.

01:56 They did tell us that there is a cosmic ray muon with a charge that is the same as an electron.

02:04 So that charge is 1 .6 times 10 to the negative 19th coulum.

02:09 And it has a mass of 1 .9 times 10 to the negative 28th kilograms.

02:17 And finally, it is saying that it is moving directly downward towards the earth's surface with a speed of 4 .5 times.

02:27 Times 10 to the seventh meters per second.

02:31 And if i kind of draw the velocity vector in relation to what i've done here, remember i said that this is the horizontal, which you can see as the earth's surface.

02:42 And so if you're coming straight down at it, then you can draw the velocity vector like so.

02:54 Okay? and the question is asking you, what is the magnitude and direction of the magnetic field? so we know that for magnetic field, what matters is q times b crossed with b, which is either q times the horizontal component of velocity relative to the magnetic field or when you have velocity times the horizontal component of the magnetic field relative to the velocity vector.

03:24 So let's see which one makes a bit more sense based on our diagram.

03:28 So notice that you can actually split the magnetic field into a component parallel to the velocity and a component perpendicular to the velocity.

03:42 Right.

03:43 And because this is 55, this angle right here is also 55.

03:48 So we're used to writing the equation for magnetic field as q times b times b times the angle between q and b.

03:58 Sorry, between b and b...

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