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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

In 1897, J. J. Thomson "discovered" the electron by measuring the

charge-to-mass ratio of "cathode rays" (actually, streams of electrons, with charge qand mass m)as follows:

(a) First he passed the beam through uniform crossed electric and magnetic fields E鈫�and B鈫�(mutually perpendicular, and both of them perpendicular to the beam), and adjusted the electric field until he got zero deflection. What, then, was the speed of the particles in terms of E鈫�and B鈫�)?

(b) Then he turned off the electric field, and measured the radius of curvature, R,

of the beam, as deflected by the magnetic field alone. In terms of E, B,and R,

what is the charge-to-mass ratio (qlm)of the particles?

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.

A current Iflows down a wire of radius a.

(a) If it is uniformly distributed over the surface, what is the surface current density K?

(b) If it is distributed in such a way that the volume current density is inversely

proportional to the distance from the axis, what is J(s)?

Find the magnetic field at point Pon the axis of a tightly woundsolenoid(helical coil) consisting of nturns per unit length wrapped around a cylindrical tube of radius aand carrying current I(Fig. 5.25). Express your answer in terms of 胃1and 胃2 (it's easiest that way). Consider the turns to be essentially circular, and use the result of Ex. 5.6. What is the field on the axis of an infinitesolenoid (infinite in both directions)?

Use the results of Ex. 5.11to find the magnetic field inside a solid sphere, of uniform charge density 蚁and radius R, that is rotating at a constant angular velocity \omega.
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