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Chapter 30: Q68P (page 900)

A toroidal inductor with an inductance of 9.0.mH encloses a volume of $0.0200 m^{3}$ . If the average energy density in the toroid is $70.0 J / m^{3}$ , what is the current through the inductor?

Short Answer

Expert verified

5.58 A

Step by step solution

01

Given

i) Inductance of inductor $L = 90.0 mH = 90.0 脳 10^{- 3} H$

ii) Volume of toroid $V = 0.0200 m^{3}$

ii) Average density $u_{B} = 70.0 J / m^{3}$ l

02

Understanding the concept

We use the formula of total energy stored for volumein the magnetic field into the formula of energy stored in the inductor鈥檚 magnetic field to find the current through the inductor.

Formula:

$U_{B} = \frac{1}{2} L i^{2} U_{B} = u_{B} V$

03

Calculate the current through the inductor

The magnetic field stored in the toroid is given by

$U_{B} = \frac{1}{2} L i^{2}$

Butthe total energy stored for volume V in the magnetic field is given by

$U_{B} = u_{B} V$

Therefore,

$\frac{1}{2} L i^{2} = u_{B} V i = \sqrt{\frac{2 u_{B} V}{L}} i = \sqrt{\frac{2 (70.0 \frac{J}{m^{3}}) (0.0200 m^{3})}{(90.0 脳 10^{- 3} H)}} i = 5.58 A$

Therefore, the current through the inductor is 5.58 A.

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Most popular questions from this chapter

Question: In Figure, a stiff wire bent into a semicircle of radius a = 2.0cmis rotated at constant angular speed $40 \frac{rev}{s}$ in a uniform 20mTmagnetic field. (a) What is the frequency? (b) What is the amplitude of the emf induced in the loop?

For the circuit of Figure, assume that $蔚 = 10.0 V, R = 6.70 惟, and L = 5.50 H$ . The ideal battery is connected at time $t = 0$ . (a) How much energy is delivered by the battery during the first 2.00 s? (b) How much of this energy is stored in the magnetic field of the inductor? (c) How much of this energy is dissipated in the resistor?

How long would it take, following the removal of the battery, for the potential difference across the resistor in an RL circuit (with L = 2.00H, R = 3.00) to decay to 10.0% of its initial value?

Two coils connected as shown in Figure separately have inductances L₁ and L₂. Their mutual inductance is M. (a) Show that this combination can be replaced by a single coil of equivalent inductance given by

$L_{eq} = L_{1} + L_{2} + 2 M$

(b) How could the coils in Figure be reconnected to yield an equivalent inductance of

$L_{eq} = L_{1} + L_{2} - 2 M$

(This problem is an extension of Problem 47, but the requirement that the coils be far apart has been removed.)

Figure 30-73a shows two concentric circular regions in which uniform magnetic fields can change. Region 1, with radius, has an outward magnetic field that is increasing in magnitude. Region 2, with radius $r_{2} = 2.0 cm$ , has an outward magnetic field that may also be changing. Imagine that a conducting ring of radius R is centered on the two regions and then the emf around the ring is determined. Figure 30-73b gives emf as a function of the square R2 of the ring鈥檚 radius, to the outer edge of region 2. The vertical axis scale is set by $E_{s} = 20 nV$ . What are the rates (a) $\frac{{dB}_{1}}{dt}$ and (b) $\frac{{dB}_{2}}{dt}$ ? (c) Is the magnitude of increasing, decreasing, or remaining constant?

See all solutions

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