A small compass is held horizontally, the center of its needle a distance of 0.280 m directly no

A small compass is held horizontally, the center of its...

Key Concepts

Compass Functionality

A compass needle aligns itself along the net magnetic field present at its location. When additional magnetic fields are introduced, such as from a nearby current, the compass needle deviates from its usual alignment with the Earth’s magnetic field. This deviation can be used to analyze the magnitude and direction of the additional field components.

Right-Hand Rule

The right-hand rule is a method used to determine the direction of the magnetic field generated by a current. When the thumb of the right hand points in the direction of the current, the curled fingers indicate the direction of the magnetic field loops around the wire. This rule is essential in predicting the orientation of the magnetic field relative to the compass and other objects in its vicinity.

Vector Addition of Magnetic Fields

When multiple magnetic fields are present, the net magnetic field at any point is found by vectorially adding all the individual fields. This principle is used in analyzing situations where the Earth's magnetic field and another magnetic field (from a current-carrying wire, for instance) combine to alter the direction in which a compass needle points.

Magnetic Field of a Current-Carrying Conductor

A long, straight current-carrying wire produces a magnetic field in the surrounding space, the magnitude of which can be calculated by the expression B = ??I/(2?r), where I is the current and r is the perpendicular distance from the wire. This concept is crucial in understanding how a current can alter the magnetic environment and interact with other magnetic fields.

Earth's Horizontal Magnetic Field

The Earth has a magnetic field that, at many locations, has a significant horizontal component. This horizontal component is what compasses align to, providing directional information. In problems involving magnetic interactions near the Earth's surface, it is often important to consider this horizontal component to determine how additional magnetic fields alter the net field experienced by a compass.

Transcript

00:02 Okay, so the setup, the situation that we have here is we have a compass that is held horizontally to the earth's surface, which means that if you were holding it, you would have the north pointing in basically the direction north.

00:27 So you have it held, it'd be like holding it in your palm flat above the earth, is what they mean by it's horizontally to the earth.

00:35 So we've got a compass here.

00:38 So north is pointing parallelly along the surface of the earth.

00:43 And we have a wire coming from the earth pointing up to the sky.

00:48 So in the context of the page here, it will be coming out of the page.

00:54 And this is the universal symbol for that, meaning that it's pointing straight up out of the page.

01:00 And we want to know what direction, the...

01:08 We want to find the magnitude of the earth's horizontal component.

01:14 So when there is no current in this wire, we know that the compass points straight north.

01:21 That's when i is equal to zero.

01:23 When there is a current flowing through the wire, we are told that the compass changes to an angle theta.

01:39 Run out of the room, so i'm going to write it out here.

01:44 Theta is this angle.

01:48 So we want to know what direction the current's flowing in this wire, whether it's flowing towards the earth or up towards the sky or away from the earth.

01:57 And we want to know the magnitude of the earth's magnetic field.

02:02 So first off, we're told that no current the needle points north, which means there's an external magnetic field from the earth, which is good because we need to find that in part b.

02:12 So that tells us that that is true that there is a magnetic force other than from the wire when there are a is a current in the wire, we get a changed angle here.

02:24 So by the right -hand rule, if this moves to the left, which means your fingers will curl to the left, means your thumbs pointing up.

02:33 So that means the current will be flowing out of the page like i have here already, which will cause this to move this way.

02:41 So that means that the current in the wire is flowing towards the sky or away from the earth.

02:48 Once again, if you were to point your fingers up out of the page, if you point your thumb up out of the page, your fingers curl around this way, which puts the force on this needle to move to this angle theta from the north.

03:10 So that's part a.

03:13 It's moving away from the earth.

03:16 If it were going into the page, just for reference, right -hand rule, you'd point your thumb down towards the page, and then for your right hand, your fingers would curl the other way.

03:32 They'd curl around that way.

03:36 But this is not what we want.

03:39 We have this situation.

03:41 So that's part a.

03:42 For part b, we have our equation for magnetic field from a current carrying wire b, is equal to mu not the constant times the current divided by 2 pi times the radius.

04:16 So all of our known values, things that were given in the problem.

04:20 We have the current flowing through the wire, and we have the distance from the wire here to our needle, d.

04:38 Or in this case, we can call it r, because that's where our equation.

04:40 Calls for so we're given that distance and we are also given the angle theta but that doesn't play into this equation that's something we'll have to do later on so we have everything we need here to solve for the magnetic field at the distance d here from the needle so plugging all that in we get our number b which is the magnetic field the distance d from the wire to the needle that's our magnetic field there now from the right -hand rule that tells us at that point the magnetic field points in the direction straight to the right.

05:40 Basically any point along the axis straight from the wire, your fingers will hit it at a completely 90 degree angle and point to the right, which is where our needle is.

05:55 So at that point it points straight to the right.

05:58 But we have an angle.

06:04 We also know that with no current, the force from the earth is pointing directly north...

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