At New York City, the earth's magnetic field has a vertical component of 5.2 ×10^-5 T that point

step 1 Okay, this is an interesting problem. We are given two magnetic fields that are perpendicular to

At New York City, the earth's magnetic field has a vertical...

Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept involves the force experienced by a wire carrying an electric current when it is placed in a magnetic field. The force is described by the equation F = I L × B, where I is the current, L is the vector representing the length and direction of the wire, and B is the magnetic field vector. This principle is a fundamental aspect of electromagnetism and is essential for understanding how current-carrying conductors interact with magnetic fields in devices like motors and generators.

Vector Cross Product

The vector cross product is a mathematical operation used to determine a vector that is perpendicular to the two input vectors. In the context of the magnetic force, the cross product between the current vector (related to the wire’s direction and magnitude) and the magnetic field vector results in the force vector. The magnitude of this force is given by the product of the magnitudes of the two vectors and the sine of the angle between them, while its direction is determined by the right-hand rule.

Magnetic Field Decomposition

This concept refers to the process of breaking down a magnetic field into its component vectors, typically along orthogonal axes such as horizontal and vertical directions. Decomposing a magnetic field allows for the analysis of its effects on charges or current-carrying conductors in different orientations. Understanding these components is important when evaluating forces in environments where the magnetic field is not uniform or is oriented at an angle relative to the conductor.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the vector resulting from the cross product in electromagnetic contexts. When applying this rule to a current-carrying conductor in a magnetic field, you point your thumb in the direction of the current and your fingers in the direction of the magnetic field; the direction your palm pushes corresponds to the force on the wire. This rule is essential for visualizing and predicting the orientation of forces in magnetic interactions.

Transcript

00:02 Okay, this is an interesting problem.

00:05 We are given two magnetic fields that are perpendicular to each other.

00:09 So let's start.

00:10 We know that this is going to be a three -dimensional problem.

00:14 So let's draw three -dimensional axis here.

00:16 So we have our z axis.

00:18 We have our x -axis.

00:20 We have our y -axis.

00:22 So now we have negative z in this direction.

00:26 We have negative x in this direction, and then our negative y goes this way.

00:34 So now we have two magnetic forces, our magnetic fields.

00:40 Let's do those in green.

00:41 Magnetic field.

00:43 We have one flowing into the earth.

00:48 Now, for simplicity, let's call the, this will be x.

00:53 Let's call the x, y plane.

00:56 So the plane created by these two, this will be our earth.

01:06 And z will be into and or out of the earth.

01:10 So we have one component of the magnetic field flowing into the earth or in the negative z direction, and then we have another portion of the magnetic field flowing in the negative x direction, which is parallel to the earth, flowing this way.

01:28 Okay, so we have that.

01:32 Let's draw, i'm going to take this xy plane and move it over here, just so we can play around with it and see, have a better description of what we're doing.

01:53 Drawing in three dimensions is always fun.

01:56 So there's our plane.

01:58 We are told we have a wire that flows straight down into the earth.

02:05 That's the direction of flow.

02:07 We don't care where it's going or where it came from.

02:10 We only care about this specific length of layer, we are given.

02:15 We hold it has a charge flow.

02:18 So we're told the length and we know the current flowing in the wire.

02:23 And current is typically represented by the letter i, capital i.

02:29 And we are also given both of these two magnetic fields.

02:33 I'm going to call this b sub z, the b direction, and this one, b sub y, or not it's b sub x for the x direction.

02:43 B sub x direction.

02:48 V sub x.

02:50 Okay, so now what equation are we going to use? we have a perfect equation that represents all of the factors that we're given here.

03:02 The force, which is what we're trying to find here, we want the magnitude and the direction of the force.

03:09 So we're going to want to keep our directions in mind while we're calculating this, which means we're probably going to have to use the right -hand rule.

03:16 So we have the force is equal to the current.

03:22 Now this equation relates magnetic field to current or a flowing charge.

03:28 So it's the current times the length times the magnetic field.

03:34 Now the important factor here is that it's sine of theta of all of that, all of that multiplied by sine of theta.

03:42 Which, what does that mean exactly? so typically when you have a sign in your equation, it's taking into consideration only the portion of this segment that relates to this segment in the perpendicular fashion.

04:04 So the current flowing only perpendicular to be will be considered here...

Please give Ace some feedback