The x, y, and z components of a magnetic field are Bx=0.10 T By=0.15 T , and Bz=0.17 T . A 25-cm

The x, y, and z components of a magnetic field are Bx=0.10 T...

The $x, y,$ and $z$ components of a magnetic field are $B_{x}=0.10 \mathrm{T}$ $B_{y}=0.15 \mathrm{T}$ , and $B_{z}=0.17 \mathrm{T} . \mathrm{A} 25-\mathrm{cm}$ wire is oriented along the $z$ axis and carries a current of 4.3 $\mathrm{A}$ . What is the magnitude of the magnetic force that acts on this wire?

Key Concepts

Cross Product and Right-Hand Rule

The cross product is a vector operation that yields a vector perpendicular to the plane containing two input vectors. In the context of magnetic forces, the cross product determines both the magnitude (using the product of the magnitudes and the sine of the angle between them) and the direction of the magnetic force, with the direction given by the right-hand rule.

Magnetic Field

A magnetic field is a vector field that describes the force that would be exerted on moving electric charges and magnetic dipoles at various points in space. It is characterized by both magnitude and direction, and in many problems is broken down into components along a coordinate system to simplify analysis.

Current-Carrying Conductor

A current-carrying conductor consists of materials through which an electric current flows, creating a magnetic field around it. When placed in an external magnetic field, the conductor experiences a force dependent on the current, the conductor's length, and its orientation relative to the magnetic field.

Magnetic Force on a Conductor (Lorentz Force)

The magnetic force on a current-carrying wire is described by the formula F = I (L × B), where I is the current, L is the vector representing the wire's length and orientation, and B is the magnetic field. This force is perpendicular to both the current direction and the magnetic field, and its magnitude depends on the sine of the angle between L and B.

Transcript

00:01 In this problem, we're given information about the components of a magnetic field vector at a particular location.

00:08 And then we're said that some wire of a particular length is oriented along one of our coordinate directions, and that wire isn't going to carry a particular current.

00:17 And so we just want to figure out what is the magnetic force acting on that particular wire.

00:22 And really, we just want the magnitude of that force.

00:24 So that's nice.

00:24 One less thing to look at.

00:25 So let's just get an idea of what this situation looks like.

00:29 Let's draw a coordinate.

00:30 System.

00:31 So we'll have our z direction heading this way.

00:33 Let's put our x direction this way and our y direction along this way.

00:41 So at this particular, say, location, we'll put it at the origin.

00:44 We have information about the components of the magnetic field at that location.

00:50 Right.

00:50 So we know that along the x axis, we're going to have a magnetic field component along the x axis that's given by, in this case, .1 teslas.

01:00 We're going to have a component in the y direction that's given by, let's try to draw a little longer, right? b -y, 0 .15 teslas.

01:11 And then we have a component in the z direction, a little bit longer than that, just a tad, bz is equal to 0 .17 tesla.

01:21 Okay, so that's the information about the components of the magnetic field.

01:25 We're also told that there is a wire that runs along the z direction, say something like this, and it carries a current i in it, and that current is going to be, let's see, given as 4 .3 amps.

01:49 Cool.

01:49 We also know that the wire has a particular length.

01:52 We are given that length.

01:53 Let me draw a little better...

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