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Chapter 5: Q4P (page 222)

Suppose that the magnetic field in some region has the form

B→=kzx⏜

(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→=kzx⏜ is Ia2kz⏜.

Step by step solution

01

Given data

There is a magnetic field of the formB→=kzx⏜ .

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→=||→×B→......(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→1=IaB\z-a/2z⏜=Iaka2z⏜=Ia2k2z⏜

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

F→2=IaB\z-a/2-z⏜=Iak-a2-z⏜=Ia2k2z⏜

Thus, the net force on the wire is Ia2kz⏜ .

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Most popular questions from this chapter

Question: (a) Find the magnetic field at the center of a square loop, which carries a steady current I.Let Rbe the distance from center to side (Fig. 5.22).

(b) Find the field at the center of a regular n-sided polygon, carrying a steady current

I.Again, let Rbe the distance from the center to any side.

(c) Check that your formula reduces to the field at the center of a circular loop, in

the limit n→∞.

A uniformly charged solid sphere of radius Rcarries a total charge Q, and is set spinning with angular velocitywabout the zaxis.

(a) What is the magnetic dipole moment of the sphere?

(b) Find the average magnetic field within the sphere (see Prob. 5.59).

(c) Find the approximate vector potential at a point (r,B)where r>R.

(d) Find the exact potential at a point (r,B)outside the sphere, and check that it is consistent with (c). [Hint: refer to Ex. 5.11.]

(e) Find the magnetic field at a point (r, B) inside the sphere (Prob. 5.30), and check that it is consistent with (b).

Find the exact magnetic field a distancez above the center of a square loop of side w, carrying a current. Verify that it reduces to the field of a dipole, with the appropriate dipole moment, whenz≫w.

A uniformly charged solid sphere of radius R carries a total charge Q, and is set spinning with angular velocity w about the z axis.

(a) What is the magnetic dipole moment of the sphere?

(b) Find the average magnetic field within the sphere (see Prob. 5.59).

(c) Find the approximate vector potential at a point (r, B) where r>> R.

(d) Find the exact potential at a point (r, B) outside the sphere, and check that it is consistent with (c). [Hint: refer to Ex. 5.11.]

(e) Find the magnetic field at a point (r, B) inside the sphere (Prob. 5.30), and check that it is consistent with (b).

Magnetostatics treats the "source current" (the one that sets up the field) and the "recipient current" (the one that experiences the force) so asymmetrically that it is by no means obvious that the magnetic force between two current loops is consistent with Newton's third law. Show, starting with the Biot-Savart law (Eq. 5.34) and the Lorentz force law (Eq. 5.16), that the force on loop 2 due to loop 1 (Fig. 5.61) can be written as

F2=μ04πl1l2∮∮r^r2dl1‐dl2

Figure 5.60

Figure 5.61

In this form, it is clear that F2=-F1, since role="math" localid="1657622030111" r^changes direction when the roles of 1 and 2 are interchanged. (If you seem to be getting an "extra" term, it will help to note thatdl2‐r^=dr.)
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