Force on a Current Loop in a Nonuniform Magnetic Field. It...

Force on a Current Loop in a Nonuniform Magnetic Field. It was shown in Section 27.7 that the net force on a current loop in a uniform magnetic field is zero. But what if $\overrightarrow{\boldsymbol{B}}$ is not uniform? Figure $\mathbf{P} 27.7 \mathbf{1}$ shows a square loop of wire that lies in the $x y$ -plane. The loop has corners at $(0,0),(0, L),(L, 0),$ and $(L, L)$ and carries a constant current $I$ in the clock-wise direction. The magnetic field has no $x$ -component but has both $y-$ and $z$ -components: $\overrightarrow{\boldsymbol{B}}=\left(B_{0} z / L\right) \hat{\jmath}+\left(B_{0} y / L\right) \hat{\boldsymbol{k}},$ where $B_{0}$ is a positive constant. (a) Sketch the magnetic field lines in the $y z$ -plane. (b) Find the magnitude and direction of the magnetic force exerted on each of the sides of the loop by integrating Eq. (27.20). (c) Find the magnitude and direction of the net magnetic force on the loop.

Key Concepts

Current Loop Behavior in Magnetic Fields

Current loops are basic elements in electromagnetism that illustrate how magnetic dipole moments are created. In a uniform magnetic field, symmetry ensures that the net force on a current loop typically cancels out though a torque may exist. However, in a nonuniform magnetic field, the asymmetry in the field’s distribution results in differential forces on different segments of the loop, yielding a net force. This behavior is fundamental in understanding the operation of devices that rely on magnetic levitation or magnetic sorting based on dipole interactions.

Integration of Differential Forces

When dealing with extended current-carrying structures in spatially varying magnetic fields, one must integrate the differential force contributions along the entire length of the conductor. This process is used to sum up small vector contributions from each infinitesimal segment where the magnetic field, possibly varying from point to point, interacts with the current element. The technique is essential for deriving the total magnetic force or torque acting on an object with a specific geometry.

Nonuniform Magnetic Field

A nonuniform magnetic field is one whose magnitude and/or direction changes with position. In contrast to a uniform field, where forces on symmetric arrangements like current loops can cancel, nonuniform fields exert different forces on different parts of an object. This gradient in the magnetic field can produce net forces or torques on current loops, magnetic dipoles, or ferromagnetic materials, leading to complex behavior that is important in applications like magnetic trapping or separation.

Magnetic Force on a Current-Carrying Conductor

This concept refers to the fundamental interaction between a current-carrying wire and a magnetic field, where the force on a small segment of the conductor is given by the cross product of the current element and the magnetic field. The expression dF = I (dL × B) underlies many electromagnetic phenomena and is central to devices such as motors and generators. It emphasizes the directional nature of the force and how it depends on both the current direction and the orientation of the magnetic field.

Transcript

00:01 So we're going to find the force on a current loop that's sitting in a non -uniform magnetic field.

00:10 The idea being that that force tends to always find up to be a net of zero going all the way around a loop.

00:20 If you are sitting in a uniform magnetic field, so we want to see if that is true when we are in a non -uniform field.

00:29 So what we'll need, again, like usual, is the expression f equals i -l -cross -b, except we have to integrate the length of wire over the extent of the non -uniform field, because that field could be contributing a different amount of force all the way along its length.

01:00 So they've asked us to sketch the vector field, which basically looks proportional to zy hat plus yz hat.

01:11 So it's important to note that the important part of this problem is probably not the sketch, but it is good to note that we have to have the divergence of b equal to zero.

01:25 So, in other words, magnetic fields have a lot of curl to them.

01:32 And indeed, this is true because divergence is basically the partial derivative of b x with respect to x, plus the partial of b.

01:47 Y with respect to y, plus the partial of bz with respect to z.

01:54 And all of those terms are identically zero.

01:59 So this does look like a valid non -uniform magnetic field.

02:04 What i've done with a sketch is i've basically taken zy hat plus yz hat, and i've looked at about three points along the axes of positive and negative.

02:27 And just plugged in points and saw what that vector looked like, and also chose the easy to graph areas where z was equal to y.

02:38 You just get sort of a line type of arrangement going out of 45 degrees, same with the three other quadrants.

02:49 But stepping back, you can see that there's as much of the arrows, for example, going into the origin is going out.

02:59 There's much of the arrows going up away from the axis is going down.

03:04 And that's kind of an indicator that the divergence of that magnetic field is zero.

03:11 Okay, so i'm going to break this loop up into four segments.

03:15 That's usually a good idea when you're finding four different pieces.

03:23 So we'll call them one, two, three, and four.

03:30 And really what we have is in the xy plane where z equals zero, our b is going to simplify down to b0y over l z hat.

03:49 So that's a simplification, and we can see what that simplified magnetic field does to our force.

03:56 So we'll take it one segment at a time.

03:59 The first segment is probably the easiest because along that segment we have y equals to zero...

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