PHYSICS 1250 PACKAGE >CI<
PHYSICS 1250 PACKAGE >CI<
9th Edition
ISBN: 9781305000988
Author: SERWAY
Publisher: CENGAGE LEARNING (CUSTOM)
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Chapter 31, Problem 31.80CP

An induction furnace uses electromagnetic induction to produce eddy currents in a conductor, thereby raising the conductor’s temperature. Commercial units operate at frequencies ranging from 60 Hz to about 1 MHz and deliver powers from a few watts to several megawatts. Induction heating can be used for warming a metal pan on a kitchen stove. It can be used to avoid oxidation and contamination of the metal when welding in a vacuum enclosure. To explore induction heating, consider a flat conducting disk of radius R, thickness b, and resistivity ρ. A sinusoidal magnetic field Bmax cos ωt is applied perpendicular to the disk. Assume the eddy currents occur in circles concentric with the disk. (a) Calculate the average power delivered to the disk. (b) What If? By what factor does the power change when the amplitude of the field doubles? (c) When the frequency doubles? (d) When the radius of the disk doubles?

(a)

Expert Solution
Check Mark
To determine
The average power delivered to the disk.

Answer to Problem 31.80CP

The average power delivered to the disk is πR4ω2B2maxb16ρ .

Explanation of Solution

Given info: Radius of disk is R , thickness of disk is b , resistivity of the disk is ρ and sinusoidal magnetic field is Bmaxcosωt .

Since the eddy currents occur as concentric circles with the disk. Consider the disk to be a collection of rings that each has an induced emf.

The emf induced in the disk can be given as,

ε=d(BA)dt

Here,

ε is the emf induced in the disk.

B is the magnetic field.

A is the area of the disk.

Substitute Bmaxcosωt for B and πr2 for A in the above equation,

ε=d(Bmaxcosωt)(πr2)dt=πr2ωBmaxsinωt

Here,

Bmax is the maximum magnetic field.

ω is the angular velocity.

r is the radius of the disk.

t is the time period of the disk.

The elemental resistance around the ring can be given as,

dR=ρlringAring

Here,

R is resistance in the ring

ρ is the resistivity of the disk

lring is the length of the elemental ring

Aring is the area of the elemental ring

Substitute 2πr for lring and bdr for Aring in the above equation,

dR=ρ(2πr)bdr

The power delivered to the elemental ring can be given as,

dP=ε2dR

P is the power delivered to the ring,

Substitute ρ(2πr)bdr for dR and πr2ωBmaxsinωt for ε in the above equation,

dP=(πr2ωBmaxsinωt)2(ρ(2πr)bdr)=πr3ω2B2maxsin2(ωt)bdr2ρ

The total power delivered to the disk can be given as,

P=dP

Substitute πr3ω2B2maxsin2(ωt)bdr2ρ for dP in the above equation,

P=0Rπr3ω2B2maxsin2(ωt)bdr2ρ=πω2B2maxbsin2(ωt)2ρ0Rr3dr=πR4ω2B2maxsin2(ωt)bdr8ρ

Substitute 12 for sin2(ωt) to get average power in the above equation,

Pavg=πR4ω2B2maxb16ρ (1)

Here,

Pavg is the average power delivered to the disk.

Thus, the average power delivered to the disk can be given as πR4ω2B2maxb16ρ .

Conclusion:

Therefore, the average power delivered to the disk can be given as πR4ω2B2maxb16ρ .

(b)

Expert Solution
Check Mark
To determine
The factor by which power will change when the field doubles.

Answer to Problem 31.80CP

Answer The factor by which power will change when the field doubles is four times.

Explanation of Solution

Given info: Radius of disk is R , thickness of disk is b , resistivity of the disk is ρ and sinusoidal magnetic field is Bmaxcosωt .

Explanation:

The relation between the field and the power can be given from equation (1) as,

PavgB2max

Substitute 2Bmax for Bmax in the above equation,

Pavg(2B)2max4B2max4Pavg

Thus, the power will change by four times when the field doubles.

Conclusion:

Therefore, the factor by which power will change when the field doubles is four times.

(c)

Expert Solution
Check Mark
To determine
The factor by which power will change when the frequency doubles.

Answer to Problem 31.80CP

Answer The factor by which power will change when the frequency doubles is four times.

Explanation of Solution

Given info: Radius of disk is R , thickness of disk is b , resistivity of the disk is ρ and sinusoidal magnetic field is Bmaxcosωt .

Explanation:

The relation between the field and the power can be given from equation (1) as,

Pavgω2

Substitute 2πf for ω in the above equation,

Pavg(2πf)2f2

Here,

f is the frequency of the disk.

Substitute 2f for f in the above equation,

Pavg(2f)24f24Pavg

Thus, the power will change by four times when the frequency doubles.

Conclusion:

Therefore, the factor by which power will change when the frequency doubles is four times.

(d)

Expert Solution
Check Mark
To determine
The factor by which power will change when the radius of the disk doubles.

Answer to Problem 31.80CP

Answer The factor by which power will change when the radius of the disk doubles is sixteen times.

Explanation of Solution

Given info: Radius of disk is R , thickness of disk is b , resistivity of the disk is ρ and sinusoidal magnetic field is Bmaxcosωt .

Explanation:

The relation between the field and the power can be given from equation (1) as,

PavgR4

Substitute 2R for R in the above equation,

Pavg(2R)416R416Pavg

Thus, the power will change by sixteen times when the radius of disk doubles.

Conclusion:

Therefore, the factor by which power will change when the radius of disk doubles is sixteen times.

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Chapter 31 Solutions

PHYSICS 1250 PACKAGE >CI<

Ch. 31 - The bar in Figure OQ31.6 moves on rails to the...Ch. 31 - A bar magnet is held in a vertical orientation...Ch. 31 - What happens to the amplitude of the induced emf...Ch. 31 - Two coils are placed near each other as shown in...Ch. 31 - A circuit consists of a conducting movable bar and...Ch. 31 - Two rectangular loops of wire lie in the same...Ch. 31 - In Section 7.7, we defined conservative and...Ch. 31 - A spacecraft orbiting the Earth has a coil of wire...Ch. 31 - In a hydroelectric dam, how is energy produced...Ch. 31 - A bar magnet is dropped toward a conducting ring...Ch. 31 - A circular loop of wire is located in a uniform...Ch. 31 - A piece of aluminum is dropped vertically downward...Ch. 31 - Prob. 31.7CQCh. 31 - When the switch in Figure CQ31.8a is closed, a...Ch. 31 - Prob. 31.9CQCh. 31 - A loop of wire is moving near a long, straight...Ch. 31 - A flat loop of wire consisting of a single turn of...Ch. 31 - An instrument based on induced emf has been used...Ch. 31 - Transcranial magnetic stimulation (TMS) is a...Ch. 31 - A 25-turn circular coil of wire has diameter 1.00...Ch. 31 - A circular loop of wire of radius 12.0 cm is...Ch. 31 - A circular loop of wire of radius 12.0 cm is...Ch. 31 - Prob. 31.7PCh. 31 - A strong electromagnet produces a uniform magnetic...Ch. 31 - A 30-turn circular coil of radius 4.00 cm and...Ch. 31 - Scientific work is currently under way to...Ch. 31 - An aluminum ring of radius r1 = 5.00 cm and...Ch. 31 - An aluminum ring of radius r1 and resistance R is...Ch. 31 - Prob. 31.13PCh. 31 - A coil of 15 turns and radius 10.0 cm surrounds a...Ch. 31 - A square, single-turn wire loop = 1.00 cm on a...Ch. 31 - A long solenoid has n = 400 turns per meter and...Ch. 31 - A coil formed by wrapping 50 turns of wire in the...Ch. 31 - When a wire carries an AC current with a known...Ch. 31 - A toroid having a rectangular cross section (a =...Ch. 31 - Prob. 31.20PCh. 31 - A helicopter (Fig. P30.11) has blades of length...Ch. 31 - Use Lenzs law 10 answer the following questions...Ch. 31 - A truck is carrying a steel beam of length 15.0 in...Ch. 31 - A small airplane with a wingspan of 14.0 m is...Ch. 31 - A 2.00-m length of wire is held in an eastwest...Ch. 31 - Prob. 31.26PCh. 31 - Figure P31.26 shows a lop view of a bar that can...Ch. 31 - A metal rod of mass m slides without friction...Ch. 31 - A conducting rod of length moves on two...Ch. 31 - Prob. 31.30PCh. 31 - Review. Figure P31.31 shows a bar of mass m =...Ch. 31 - Review. Figure P31.31 shows a bar of mass m that...Ch. 31 - The homopolar generator, also called the Faraday...Ch. 31 - Prob. 31.34PCh. 31 - Review. Alter removing one string while...Ch. 31 - A rectangular coil with resistance R has N turns,...Ch. 31 - Prob. 31.37PCh. 31 - An astronaut is connected to her spacecraft by a...Ch. 31 - Within the green dashed circle show in Figure...Ch. 31 - Prob. 31.40PCh. 31 - Prob. 31.41PCh. 31 - 100-turn square coil of side 20.0 cm rotates about...Ch. 31 - Prob. 31.43PCh. 31 - Figure P30.24 (page 820) is a graph of the induced...Ch. 31 - In a 250-turn automobile alternator, the magnetic...Ch. 31 - In Figure P30.26, a semicircular conductor of...Ch. 31 - A long solenoid, with its axis along the x axis,...Ch. 31 - A motor in normal operation carries a direct...Ch. 31 - The rotating loop in an AC generator is a square...Ch. 31 - Prob. 31.50PCh. 31 - Prob. 31.51APCh. 31 - Suppose you wrap wire onto the core from a roll of...Ch. 31 - A circular coil enclosing an area of 100 cm2 is...Ch. 31 - A circular loop of wire of resistance R = 0.500 ...Ch. 31 - A rectangular loop of area A = 0.160 m2 is placed...Ch. 31 - A rectangular loop of area A is placed in a region...Ch. 31 - Strong magnetic fields are used in such medical...Ch. 31 - Consider the apparatus shown in Figure P30.32: a...Ch. 31 - A guitars steel string vibrates (see Fig. 30.5)....Ch. 31 - Why is the following situation impossible? A...Ch. 31 - The circuit in Figure P3 1.61 is located in a...Ch. 31 - Magnetic field values are often determined by...Ch. 31 - A conducting rod of length = 35.0 cm is free to...Ch. 31 - Review. A particle with a mass of 2.00 1016 kg...Ch. 31 - The plane of a square loop of wire with edge...Ch. 31 - In Figure P30.38, the rolling axle, 1.50 m long,...Ch. 31 - Figure P30.39 shows a stationary conductor whose...Ch. 31 - Prob. 31.68APCh. 31 - A small, circular washer of radius a = 0.500 cm is...Ch. 31 - Figure P30.41 shows a compact, circular coil with...Ch. 31 - Prob. 31.71APCh. 31 - Review. In Figure P30.42, a uniform magnetic field...Ch. 31 - An N-turn square coil with side and resistance R...Ch. 31 - A conducting rod of length moves with velocity v...Ch. 31 - The magnetic flux through a metal ring varies with...Ch. 31 - A rectangular loop of dimensions and w moves with...Ch. 31 - A long, straight wire carries a current given by I...Ch. 31 - A thin wire = 30.0 cm long is held parallel to...Ch. 31 - Prob. 31.79CPCh. 31 - An induction furnace uses electromagnetic...Ch. 31 - Prob. 31.81CPCh. 31 - A betatron is a device that accelerates electrons...Ch. 31 - Review. The bar of mass m in Figure P30.51 is...
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What is Electromagnetic Induction? | Faraday's Laws and Lenz Law | iKen | iKen Edu | iKen App; Author: Iken Edu;https://www.youtube.com/watch?v=3HyORmBip-w;License: Standard YouTube License, CC-BY