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Chapter 20: Problem 67

A coil has an inductance of \(0.15 \mathrm{H}\) and a resistance of 33 \(\Omega\). The coil is connected to a 6.0 - \(V\) battery. After a long time elapses, the current in the coil is no longer changing. (a) What is the current in the coil? (b) What is the energy stored in the coil? (c) What is the rate of energy dissipation in the coil? (d) What is the induced emf in the coil?

Short Answer

Expert verified
Answer: The current in the coil is (2/11) A, the energy stored in the coil is (3/121) J, the rate of energy dissipation is (4/11) W, and the induced emf is 0 V.

Step by step solution

01

(a) Find the current in the coil

After a long time, the current in the coil no longer changes, and the system reaches a steady-state. In this case, the induced emf becomes zero, and the coil behaves like a regular resistor. Therefore, we can use Ohm's law to find the current in the coil: \(I = \dfrac{V}{R} = \dfrac{6 \, \text{V}}{33 \, \Omega} = \dfrac{2}{11} \, \text{A}\) So, the current in the coil is \(\dfrac{2}{11} \, \text{A}\).
02

(b) Calculate the energy stored in the coil

Using the given inductance and the calculated current, we can find the energy stored in the coil as follows: \(U = \dfrac{1}{2}LI^2 = \dfrac{1}{2}(0.15 \, \text{H})\left(\dfrac{2}{11} \, \text{A}\right)^2 = \dfrac{3}{121} \, \text{J}\) So, the energy stored in the coil is \(\dfrac{3}{121} \, \text{J}\).
03

(c) Find the rate of energy dissipation in the coil

To find the rate of energy dissipation, we use the following equation: \(P = I^2R = \left(\dfrac{2}{11} \, \text{A}\right)^2(33 \, \Omega) = \dfrac{4}{11} \, \text{W}\) So, the rate of energy dissipation in the coil is \(\dfrac{4}{11} \, \text{W}\).
04

(d) Calculate the induced emf in the coil

Since the coil has reached a steady-state and the current is no longer changing, there is no induced emf in the coil, which can be shown as: \(E = -L \dfrac{dI}{dt} = -0.15 \, \text{H} \times 0 = 0 \, \text{V}\) So, the induced emf in the coil is 0 V.

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Most popular questions from this chapter

The largest constant magnetic field achieved in the laboratory is about $40 \mathrm{T}$. (a) What is the magnetic energy density due to this field? (b) What magnitude electric field would have an equal energy density?
A solenoid of length \(2.8 \mathrm{cm}\) and diameter \(0.75 \mathrm{cm}\) is wound with 160 turns per cm. When the current through the solenoid is $0.20 \mathrm{A},$ what is the magnetic flux through one of the windings of the solenoid?
A \(0.30-\mathrm{H}\) inductor and a \(200.0-\Omega\) resistor are connected in series to a \(9.0-\mathrm{V}\) battery. (a) What is the maximum current that flows in the circuit? (b) How long after connecting the battery does the current reach half its maximum value? (c) When the current is half its maximum value, find the energy stored in the inductor, the rate at which energy is being stored in the inductor, and the rate at which energy is dissipated in the resistor. (d) Redo parts (a) and (b) if, instead of being negligibly small, the internal resistances of the inductor and battery are \(75 \Omega\) and \(20.0 \Omega\) respectively.
A 100 -turn coil with a radius of \(10.0 \mathrm{cm}\) is mounted so the coil's axis can be oriented in any horizontal direction. Initially the axis is oriented so the magnetic flux from Earth's field is maximized. If the coil's axis is rotated through \(90.0^{\circ}\) in \(0.080 \mathrm{s},\) an emf of $0.687 \mathrm{mV}$ is induced in the coil. (a) What is the magnitude of the horizontal component of Earth's magnetic field at this location? (b) If the coil is moved to another place on Earth and the measurement is repeated, will the result be the same?
An ideal inductor of inductance \(L\) is connected to an ac power supply, which provides an emf \(\mathscr{E}(t)=\mathscr{E}_{\mathrm{m}}\) sin \(\omega t\) (a) Write an expression for the current in the inductor as a function of time. [Hint: See Eq. \((20-7) .]\) (b) What is the ratio of the maximum emf to the maximum current? This ratio is called the reactance. (c) Do the maximum emf and maximum current occur at the same time? If not, how much time separates them?
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