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Chapter 7: Q13P (page 316)

A square loop of wire, with sides of length a , lies in the first quadrant of the xy plane, with one comer at the origin. In this region, there is a nonuniform time-dependent magnetic field B(y,t)=ky3t2z^ (where k is a constant). Find the emf induced in the loop.

Short Answer

Expert verified

The induced emf into loop is -kt2a5.

Step by step solution

01

Write the given data from the question.

The side of the square loop is a.

The square loop exists in xy plane.

The non-uniform time dependent magnetic field,B(y,t)=ky3t2z^

02

Calculate the induced emf in the loop.

Let鈥檚 take small strip dy on the square loop which at distance y from the x-axis.


The magnetic flux is given by,

蠒=B(y,t).dA

Here dA is the area of the strip.

Substitute Ky3t2for B(y,t)and ady for dA into above equation.

蠒=ky3t2.ady

Integrate the above equation to calculate the flus through the entire loop.

蠒=鈭�0aky3t2.ady蠒=kat2鈭�0ay3dy蠒=kat2y440a

Apply the limits,

蠒=kat2a44-044蠒=kat2a44蠒=kt24a5

The induced emf in the loop is given by,

e=-d蠒dt

Substitute kt24a5for 蠒into above equation.

e=-ddtkt24a5e=-2kt4a5e=-kt2a5

Hence the induced emf into loop is -kt2a5.

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Most popular questions from this chapter

Refer to Prob. 7.16, to which the correct answer was

E(s,t)=渭0I0蝇2蟿蟿sin(蝇t)In(as)z^

(a) Find the displacement current density Jd路

(b) Integrate it to get the total displacement current,

Id=鈭�Jd.da

Compare Id and I. (What's their ratio?) If the outer cylinder were, say, 2 mm in diameter, how high would the frequency have to be, forId to be 1% of I ? [This problem is designed to indicate why Faraday never discovered displacement currents, and why it is ordinarily safe to ignore them unless the frequency is extremely high.]

A perfectly conducting spherical shell of radius rotates about the z axis with angular velocity 蝇, in a uniform magnetic field B=B0Z^. Calculate the emf developed between the 鈥渘orth pole鈥� and the equator. Answer:localid="1658295408106" [12B0蝇伪2].

Two coils are wrapped around a cylindrical form in such a way that the same flux passes through every turn of both coils. (In practice this is achieved by inserting an iron core through the cylinder; this has the effect of concentrating the flux.) The primary coil hasN1turns and the secondary hasN2(Fig. 7.57). If the current in the primary is changing, show that the emf in the secondary is given by

蔚2蔚1=N2N1(7.67)

where蔚1is the (back) emf of the primary. [This is a primitive transformer-a device for raising or lowering the emf of an alternating current source. By choosing the appropriate number of turns, any desired secondary emf can be obtained. If you think this violates the conservation of energy, study Prob. 7.58.]

Imagine a uniform magnetic field, pointing in the zdirection and filling all space (B=B0z). A positive charge is at rest, at the origin. Now somebody turns off the magnetic field, thereby inducing an electric field. In what direction does the charge move?

In the discussion of motional emf (Sect. 7.1.3) Iassumed that the wire loop (Fig. 7.10) has a resistance R; the current generated is then I=vBhR. But what if the wire is made out of perfectly conducting material, so that Ris zero? In that case, the current is limited only by the back emf associated with the self-inductanceL of the loop (which would ordinarily be negligible in comparison with IR). Show that in this regime the loop (massm ) executes simple harmonic motion, and find its frequency. [Answer: 蝇=Bh/mL].
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