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Chapter 7: Q54P (page 349)

A circular wire loop (radiusr, resistanceR) encloses a region of uniform magnetic field,B, perpendicular to its plane. The field (occupying the shaded region in Fig. 7.56) increases linearly with timeB=伪t. An ideal voltmeter (infinite internal resistance) is connected between pointsPandQ.

(a) What is the current in the loop?

(b) What does the voltmeter read? [Answer: 伪r2/2]

Short Answer

Expert verified

(a)The current in the loop isI=蟺伪r2R .

(b) The voltmeter reading is伪r22 .

Step by step solution

01

Given information

The radius of circular wire loop is,.

The resistance of circular wire loop is, .

The uniform magnetic field inside the wire loop is, .

The relation between the magnetic field and time is, .

02

Magnetic flux

The magnetic flux inside the wire loop having magnetic field and radius is given by,

03

The current in the loop

(b)

The formula for the emf generated in the loop due to magnetic flux is given by,

Solve further as:

The negative sign indicates the emf value is decreasing.

Also, the emf using Ohm鈥檚 law,

Then equating both values,

Hence, the current in the loop is .

04

Determine the voltmeter reading value

(b)

Assume a small elemental region of radius inside the given inside the given region between points P and Q.

For a circle of radius , applying Faraday鈥檚 law for a closed area, the formula for the measured emf is given by,

In polar form,

Along the line from P to Q,

and ,

Then the voltage reading between points P and Q can be calculated as,

Hence, the voltmeter reading is .

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Most popular questions from this chapter

The magnetic field outside a long straight wire carrying a steady current I is

B=渭02蟺Is蠒^

The electric field inside the wire is uniform:

E=I蚁蟺a2z^,

Where 蚁is the resistivity and a is the radius (see Exs. 7.1 and 7 .3). Question: What is the electric field outside the wire? 29 The answer depends on how you complete the circuit. Suppose the current returns along a perfectly conducting grounded coaxial cylinder of radius b (Fig. 7.52). In the region a < s < b, the potential V (s, z) satisfies Laplace's equation, with the boundary conditions

(i) V(a,z)=I蚁z蟺a2 ; (ii) V(b,z)=0

Figure 7.52

This does not suffice to determine the answer-we still need to specify boundary conditions at the two ends (though for a long wire it shouldn't matter much). In the literature, it is customary to sweep this ambiguity under the rug by simply stipulating that V (s,z) is proportional to V (s,z) = zf (s) . On this assumption:

(a) Determine (s).

(b) E (s,z).

(c) Calculate the surface charge density 蟽(z)on the wire.

[Answer: V=(-Iz蚁/蟺a2) This is a peculiar result, since Es and 蟽(z)are not independent of localid="1658816847863" z鈫�as one would certainly expect for a truly infinite wire.]

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?

Two concentric metal spherical shells, of radius a and b, respectively, are separated by weakly conducting material of conductivity蟽(Fig. 7 .4a).

(a) If they are maintained at a potential difference V, what current flows from one to the other?

(b) What is the resistance between the shells?

(c) Notice that if b>>a the outer radius (b) is irrelevant. How do you account for that? Exploit this observation to determine the current flowing between two metal spheres, each of radius a, immersed deep in the sea and held quite far apart (Fig. 7 .4b ), if the potential difference between them is V. (This arrangement can be used to measure the conductivity of sea water.)

A small loop of wire (radius a) is held a distance z above the center of a large loop (radius b ), as shown in Fig. 7.37. The planes of the two loops are parallel, and perpendicular to the common axis.

(a) Suppose current I flows in the big loop. Find the flux through the little loop. (The little loop is so small that you may consider the field of the big loop to be essentially constant.)

(b) Suppose current I flows in the little loop. Find the flux through the big loop. (The little loop is so small that you may treat it as a magnetic dipole.)

(c) Find the mutual inductances, and confirm that M12=M21 路

A battery of emf 蔚and internal resistance r is hooked up to a variable "load" resistance,R . If you want to deliver the maximum possible power to the load, what resistance R should you choose? (You can't change e and R , of course.)
See all solutions

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