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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

Two long coaxial solenoids each carry current I , but in opposite directions, as shown in Fig. 5.42. The inner solenoid (radius a) has turns per unit length, and the outer one (radius b) has .n2Find B in each of the three regions: (i) inside the inner solenoid, (ii) between them, and (iii) outside both.

Consider a planeloop of wire that carries a steady current I;we

want to calculate the magnetic field at a point in the plane. We might as well take

that point to be the origin (it could be inside or outside the loop). The shape of the

wire is given, in polar coordinates, by a specified function r(胃)(Fig. 5.62).

(a) Show that the magnitude of the field is

role="math" localid="1658927560350" B=渭0I4蟺鈭�(5.92)

(b) Test this formula by calculating the field at the center of a circular loop.

(c) The "lituus spiral" is defined by a

r(胃)=a胃鈥夆赌夆赌夆赌夆赌�0<胃鈮�2蟺

(for some constant a).Sketch this figure, and complete the loop with a straight

segment along the xaxis. What is the magnetic field at the origin?

(d) For a conic section with focus at the origin,

r(胃)=p1+别肠辞蝉胃

where pisthe semi-latus rectum (the y intercept) and eis the eccentricity (e= 0

for a circle, 0 < e< 1 for an ellipse, e= 1 for a parabola). Show that the field is

B=渭0I2pregardless of the eccentricity.

Question: Use Eq. 5.41 to obtain the magnetic field on the axis of the rotating disk in Prob. 5.37(a). Show that the dipole field (Eq. 5.88), with the dipole moment you found in Prob. 5.37, is a good approximation if z>> R.

For a configuration of charges and currents confined within a volume

V,show that

鈭�V鈬赌闯诲蟿=dp鈬赌dt

where p鈬赌is the total dipole moment.

Question: Find the magnetic field at point Pfor each of the steady current configurations shown in Fig. 5.23.
See all solutions

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