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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

Suppose

E(r,t)=14蟺蔚0qr2胃(r鈭�蠀t)r^; B(r,t)=0

(The theta function is defined in Prob. 1.46b). Show that these fields satisfy all of Maxwell's equations, and determine 蚁 and J. Describe the physical situation that gives rise to these fields.

Two tiny wire loops, with areas and , are situated a displacement apart (Fig. 7 .42). FIGURE7.42


(a) Find their mutual inductance. [Hint: Treat them as magnetic dipoles, and use Eq. 5.88.] Is your formula consistent with Eq. 7.24?

(b) Suppose a current is flowing in loop 1, and we propose to turn on a current in loop 2. How much work must be done, against the mutually induced emf, to keep the current flowing in loop 1? In light of this result, comment on Eq. 6.35.

A square loop (side a) is mounted on a vertical shaft and rotated at angular velocity 蝇 (Fig. 7.19). A uniform magnetic field B points to the right. Find the蔚tfor this alternating current generator.

Prove Alfven's theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic field lines are, as it were, "frozen" into the fluid.)

(a) Use Ohm's law, in the form of Eq. 7.2, together with Faraday's law, to prove that if 蟽=鈭�and is J finite, then

鈭�B鈭�t=鈭�脳(v脳B)

(b) Let S be the surface bounded by the loop (P)at time t , and S'a surface bounded by the loop in its new position (P')at time t+dt (see Fig. 7.58). The change in flux is

诲桅=鈭�S'B(t+dt)鈥�da-鈭�SB(t)鈥�da

Use 鈭�路B=0to show that

鈭�S'B(t+dt)鈥�da+鈭�RB(t+dt)鈥�da=鈭�SB(t+dt)鈥�da

(Where R is the "ribbon" joining P and P' ), and hence that

诲桅=dt鈭�S鈭�B鈭�t路da-鈭�RB(t+dt)鈥�da

(For infinitesimal dt ). Use the method of Sect. 7.1.3 to rewrite the second integral as

dt鈭�P(B脳v)路dI

And invoke Stokes' theorem to conclude that

诲桅dt=鈭�S(鈭�B鈭�t-鈭�脳v脳B)路da

Together with the result in (a), this proves the theorem.

A thin uniform donut, carrying charge Q and mass M, rotates about its axis as shown in Fig. 5.64.

(a) Find the ratio of its magnetic dipole moment to its angular momentum. This is called the gyromagnetic ratio (or magnetomechanical ratio).

(b) What is the gyromagnetic ratio for a uniform spinning sphere?[This requires no new calculation; simply decompose the sphere into infinitesimal rings, and apply the result of part (a).]

(c) According to quantum mechanics, the angular momentum of a spinning electron is 12鈩� , where 鈩� is Planck's constant. What, then, is the electron's magnetic dipole moment, in localid="1657713870556" Am2 ? [This semi classical value is actually off by a factor of almost exactly 2. Dirac's relativistic electron theory got the 2 right, and Feynman, Schwinger, and Tomonaga later calculated tiny further corrections. The determination of the electron's magnetic dipole moment remains the finest achievement of quantum electrodynamics, and exhibits perhaps the most stunningly precise agreement between theory and experiment in all of physics. Incidentally, the quantity(localid="1657713972487" (e鈩�/2m), where e is the charge of the electron and m is its mass, is called the Bohr magneton.]
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