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Chapter 5: Q4P (page 222)

Suppose that the magnetic field in some region has the form

B鈫�=kzx铃�

(where kis a constant). Find the force on a square loop (side a),lying in the yz

plane and centered at the origin, if it carries a current I,flowing counterclockwise,

when you look down the xaxis.

Short Answer

Expert verified

The force on a square loop of side a,lying in the yzplane and centered at the origin, carrying a current I,flowing counterclockwise in the presence of a magnetic fieldB鈫�=kzx铃� is Ia2kz铃�.

Step by step solution

01

Given data

There is a magnetic field of the formB鈫�=kzx铃� .

There is a square loop of side alying in the yzplane, centered at the origin and carries a current I,flowing counterclockwise.

02

Force on a current carrying wire in a magnetic field

The force on a wire carrying current I, length Iin a magnetic fieldBis

F鈫�=||鈫�脳B鈫�......(1)

03

Force on the current carrying loop

The force on y=a/2and y=-a/2wires are exactly equal and opposite and so they cancel out.

From equation (1), force on the z=a/2 wire

F鈫�1=IaB\z-a/2z铃�=Iaka2z铃�=Ia2k2z铃�

From equation (1), force on the z=-a/2 wire

F鈫�2=IaB\z-a/2-z铃�=Iak-a2-z铃�=Ia2k2z铃�

Thus, the net force on the wire is Ia2kz铃� .

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Most popular questions from this chapter

Suppose you wanted to find the field of a circular loop (Ex. 5.6) at a point rthat is not directly above the center (Fig. 5.60). You might as well choose your axes so that rlies in the yzplane at (0,y,z). The source point is ( Rcos 蠁',Rsin 蠒',0, and 蠒'runs from 0 to 2JJ. Set up the integrals25 from which you could calculate Bx,Byand Bzand evaluate Bxexplicitly.

It may have occurred to you that since parallel currents attract, the current within a single wire should contract into a tiny concentrated stream along the axis. Yet in practice the current typically distributes itself quite uniformly over the wire. How do you account for this? If the positive charges (density 蚁+) are "nailed down," and the negative charges (density蚁-) move at speed v(and none of these depends on the distance from the axis), show that 蚁-=-蚁+纬2,Where纬鈮�1/1-(v/c)2andc2=1/渭0蔚0. If the wire as a whole is neutral, where is the compensating charge located?22[Notice that for typical velocities (see Prob. 5.20), the two charge densities are essentially unchanged by the current (since纬鈮�1). In plasmas, however, where the positive charges are also free to move, this so-called pinch effect can be very significant.]

In calculating the current enclosed by an Amperian loop, one must,in general, evaluate an integral of the form

Ienc=鈭�sJ鈰�da

The trouble is, there are infinitely many surfaces that share the same boundary line. Which one are we supposed to use?

Is Ampere's law consistent with the general rule (Eq. 1.46) that divergence-of-curl is always zero? Show that Ampere's law cannot be valid, in general, outside magnetostatics. Is there any such "defect" in the other three Maxwell equations?

A particle of charge qenters a region of uniform magnetic field B鈫� (pointing intothe page). The field deflects the particle a distanced above the original line of flight, as shown in Fig. 5.8. Is the charge positive or negative? In terms of a, d, Band q,find the momentum of the particle.
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