In a certain region, the carth's magnetic ficld has a magnitude of 5.4 ×10^-5 T and is directed

In a certain region, the carth's magnetic ficld has a...

In a certain region, the carth's magnetic ficld has a magnitude of $5.4 \times 10^{-5} \mathrm{T}$ and is directed north at an angle of $58^{\circ}$ below the horizontal. An electrically charged bullet is fired north and $11^{\circ}$ above the horizontal, with a speed of 670 $\mathrm{m} / \mathrm{s}$ . The magnetic force on the bullet is $2.8 \times 10^{-10} \mathrm{N}$ directed due east. Determine the bullet's electric charge, including its algebraic sign $(+\text { or }-) .$

Key Concepts

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving through a magnetic field, given by F = q(v × B). This concept is fundamental to understanding the interaction between charged particles and magnetic fields, as it quantifies how the motion and the magnetic environment determine the resulting force.

Vector Cross Product

The cross product operation is crucial when dealing with the magnetic force because it provides the magnitude and direction of the force from two vectors (velocity and magnetic field) that are generally not parallel. The sine of the angle between these vectors plays a key role in determining the strength of the force.

Trigonometric Decomposition

When dealing with vectors oriented at angles relative to standard axes, trigonometric decomposition is used to break them into their horizontal and vertical components. This allows for precise calculation of the effective magnitudes and directions involved in the vector cross product, particularly when the given angles are not aligned with the coordinate axes.

Determining Charge from Force

In problems where the magnetic force is known but the charge is not, the reverse application of the Lorentz force equation enables the determination of the charge. By analyzing the magnitudes of the force, velocity, and magnetic field—including the sine of the angle between velocity and magnetic field vectors—the charge and its sign can be inferred.

Transcript

00:01 Alright, so for this problem, i think the hardest part is just trying to understand the directions that we have.

00:10 So a typical compass, just for starters, has north up top, south, east on the right, and west on the left.

00:28 So we're told we have the earth with a magnetic field, the earth's magnetic field, b.

00:40 And it's at an angle that is below the horizontal.

00:45 So what i'm going to do to represent that is turn this compass flat, with west pointing back into the page, east coming out of the page, and north heading to the right, like so.

01:00 So we have north.

01:03 Let's do it in black.

01:05 Actually, yeah, let's do it in red.

01:07 So we have north pointing to the right.

01:13 Which let's put a point here to signify our origin which means south is this way is it north south so west is going back into the page and east is coming out of the page now i'm doing this because this is a three -dimensional problem they're having us do so i'm going to draw it with the three -dimensional axis so here's the compass laid flat now we have two other dimensions, which is up and down.

02:05 So this makes a 3d graph.

02:19 So up and down.

02:22 Now we have our magnetic field is below the horizontal.

02:29 So in the perspective of the earth, that would be going down into the ground.

02:36 If you look north, that would be angled down towards the ground.

02:40 And this is at theta 1, at angle theta 1, which i'm just going to call theta 1 here.

02:49 And now we have a bullet that's fired up above the horizontal at an angle of theta 2.

03:08 Now these are in this up -north plane, the plane created between the up -axis and the north axis.

03:22 So this w axis is going back into the page.

03:26 So this would come down here and touch north.

03:29 And this would come down and touch, or this would come up and touch north.

03:41 So those are our angles that we have.

03:44 So the bullet is fired upwards towards the sky.

03:47 The magnetic field is angled downwards into the ground, both pointing north or northward.

03:56 So we have the magnetic field.

03:58 Of the earth, we have the velocity the bullet is fired at, we have the force on the bullet, and we need to find out what our overall angle is.

04:17 I'm going to call it theta t for the total angle, because we want to use the equation for force with the electric field, or magnetic field.

04:35 So our equation for that, because we want to find what the charge of the charge of the field, the bullet is.

04:39 We're told the bullet has a charge on it...

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