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Chapter 29: Problem 6

A coil 4.00 \(\mathrm{cm}\) in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to \(B=(0.0120 \mathrm{T} / \mathrm{s}) t+\left(3.00 \times 10^{-5} \mathrm{T} / \mathrm{s}^{4}\right) t^{4} .\) The coil is connected to a \(600-\Omega\) resistor, and its plane is perpendicular to the magnetic field. You can ignore the resistance of the coil. (a) Find the magnitude of the induced emf in the coil as a function of time. (b) What is the current in the resistor at time \(t=5.00 \mathrm{s} ?\)

Short Answer

Expert verified
(a) The emf is \\(-0.03015 - 0.0003014 t^3\). (b) The current at \\(t=5\\) s is \\-0.000113\\) A.

Step by step solution

01

Understand Faraday's Law of Induction

To find the induced emf, we will apply Faraday's Law of Electromagnetic Induction. According to Faraday's Law, the induced emf \( \varepsilon \) in a coil with \( N \) turns is given by: \[ \varepsilon = -N \frac{d\Phi_B}{dt} \] where \( \Phi_B \) is the magnetic flux through one loop of the coil.
02

Calculate the Magnetic Flux

The magnetic flux \( \Phi_B \) through one loop is given by: \[ \Phi_B = B \cdot A \] where \( A \) is the area of a single loop of the coil. The radius of the coil is \( 4.00 \) cm or \( 0.04 \) m, so \[ A = \pi r^2 = \pi (0.04)^2 = 5.024 \times 10^{-3} \text{ m}^2 \] Thus, the flux is: \[ \Phi_B = (0.0120 t + 3.00 \times 10^{-5} t^4) \times 5.024 \times 10^{-3} \] \[ \Phi_B = 6.029 \times 10^{-5} t + 1.507 \times 10^{-7} t^4 \]
03

Differentiate the Magnetic Flux

Differentiate the expression for magnetic flux \( \Phi_B(t) \) with respect to time \( t \): \[ \frac{d\Phi_B}{dt} = \frac{d}{dt}(6.029 \times 10^{-5} t + 1.507 \times 10^{-7} t^4) \] \[ \frac{d\Phi_B}{dt} = 6.029 \times 10^{-5} + 4 \times 1.507 \times 10^{-7} t^3 \] \[ \frac{d\Phi_B}{dt} = 6.029 \times 10^{-5} + 6.028 \times 10^{-7} t^3 \]
04

Calculate the Induced EMF

Substitute \( \frac{d\Phi_B}{dt} \) into the expression for the induced emf: \[ \varepsilon = -500 \, (6.029 \times 10^{-5} + 6.028 \times 10^{-7} t^3) \] \[ \varepsilon = -500 \times (6.029 \times 10^{-5} + 6.028 \times 10^{-7} t^3) \] \[ \varepsilon = -0.03015 - 0.0003014 t^3 \] \[ \varepsilon = -(0.03015 + 0.0003014 t^3) \]
05

Calculate the Current in the Resistor

Use Ohm's Law to find the current \( I \) in the resistor, \( I = \frac{\varepsilon}{R} \). The resistor's resistance \( R \) is given as 600 \( \Omega \): For \( t = 5 \) s: \[ \varepsilon(5) = -(0.03015 + 0.0003014 \times 5^3) \] \[ \varepsilon(5) = -(0.03015 + 0.0003014 \times 125) \] \[ \varepsilon(5) = -(0.03015 + 0.037675) \] \[ \varepsilon(5) = -0.067825 \] Thus, \[ I = \frac{-0.067825}{600} = -0.000113 \text{ A} \] The negative sign indicates the direction of the current, which is typically opposite to the assumed direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Induced emf
In the realm of electromagnetism, induced electromotive force (emf) is a significant concept that explains how electricity can be generated from a changing magnetic field. According to Faraday's Law of Induction, the induced emf in a coil with several turns is related to the rate of change of magnetic flux through the coil. Simply put, whenever there is a change in the magnetic environment of a coil of wire, it results in an induced current. This phenomenon is fundamentally explained by the formula:
  • \( \varepsilon = -N \frac{d\Phi_B}{dt} \)
Here, \( \varepsilon \) is the induced emf, \( N \) represents the number of turns in the coil, and \( \frac{d\Phi_B}{dt} \) is the rate of change of the magnetic flux.
In practice, when you have a coil placed in a magnetic field that varies over time, the change in magnetic field strength, even if marginal, will cause an emf to be induced. This is a cornerstone principle in the operation of devices like electric generators and transformers. Faraday's Law was the dawn of understanding how mechanical energy can be converted into electrical energy, an idea that underpins much of today's power generation technologies.
Magnetic flux
Magnetic flux (\( \Phi_B \)) is a measure of the quantity of magnetism, taking into account the strength and extent of a magnetic field. You could think of it as the total magnetic 'field lines' passing perpendicularly through a given area. The greater the number of lines passing through the coil, the greater the magnetic flux. Mathematically, it is given by:
  • \( \Phi_B = B \cdot A \)
Where \( B \) is the magnetic field strength and \( A \) is the area of the loop.
For example, if a coil with a certain radius is in a magnetic field, its area can be calculated using the formula for the area of a circle, \( A = \pi r^2 \). In the given exercise, substituting the coil's radius into this will help us determine how much magnetic field permeates the coil.
Understanding magnetic flux is crucial because it connects the physical size and shape of the coil with the magnetic environment it is in. By knowing the magnetic flux, one can predict how strong the induced emf will be when there are adjustments in the magnetic field, thanks to the quoted expression in Faraday鈥檚 Law.
Ohm's Law
Ohm鈥檚 Law is a fundamental principle in the analysis of electrical circuits. It connects three essential quantities: voltage (or emf), current, and resistance, providing a straightforward relationship between them. This law states that the current (\( I \)) flowing through a conductor between two points is directly proportional to the voltage (\( V \)) across the two points and inversely proportional to the resistance (\( R \)) of the conductor. It is articulated as:
  • \( I = \frac{V}{R} \)
Within the context of the given problem, once the induced emf (which acts like a voltage) is known, Ohm's Law enables us to calculate the current passing through a resistor connected to the coil. The resistance of the resistor is given as 600 ohms.
For instance, at a specific time, by substituting known values for the emf and resistance into Ohm's Law, the current can be accurately determined. This utility of Ohm's Law simplifies calculations in circuits and is pivotal in designing electronic devices, ensuring that the right amount of current flows through various components to function correctly.

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Most popular questions from this chapter

Displacement Current in a Wire. A long, straight, copper wire with a circular cross-scctional area of 2.1 \(\mathrm{mm}^{2}\) carries a current of 16 \(\mathrm{A}\) . The resistivity of the material is \(20 \times 10^{-8} \Omega \cdot \mathrm{m}\) . (a) What is the uniform electric field in the material? (b) If the cur- rent is changing at the rate of 4000 \(\mathrm{A} / \mathrm{s}\) , at what rate is the electric field in the material changing? (c) What is the displacement current density in the material in part (b)? (Hint: Since \(K\) for copper is very close to \(1,\) use \(\epsilon=\epsilon_{0} . )\) (d) If the current is changing as in part (b), what is the magnitude of the magnetic field 6.0 \(\mathrm{cm}\) from the center of the wire? Note that both the conduction current and the displacement current should be included in the calculation of \(B\) . Is the contribution from the displacement current significant?

Make a Generator? You are shipwrecked on a deserted tropical island. You have some electrical devices that you could uperate using a generalor but you have nu maguels. The eardis magnetic field at your location is horizontal and has magnitude 8.0 \(\times 10^{-5} \mathrm{T}\) , and you decide to try to use this field for a generator by rotating a large circular coil of wire at a high rate. You need to produce a peak emf of 9.0 \(\mathrm{V}\) and estimate that you can rotate the coil at 30 \(\mathrm{rpm}\) by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum mumber of turns the coil can have is 2000 . (a) What area must the coil have? (b) If the coil is circular, what is the maximum translational speed of a point on the coil as it rotates? Do you this device is feasible? Explain.

The magnetic field within a long, straight solenoid with a eireular cross section and radius \(R\) is increasing at a rate of \(d B / d t .\) (a) What is the rate of change of flux through a circle with radius \(r_{1}\) inside the solenoid, normal to the axis of the solenoid, and with center on the solenoid axis? (b) Find the magnitude of the induced electric field inside the solenoid, at a distance \(r_{1}\) from its axis. Show the direction of this field in a diagram. (c) What is the magnitude of the induced electric field outside the solenoid, at a distance \(r_{2}\) from the axis? (d) Graph the magnitude of the induced electric field as a function of the distance \(r\) from the axis from \(r=0\) to \(r=2 R\) . (e) What is the magnitude of the induced emf in circular turn of radius \(R / 2\) that has its center on the solenoid axis? (f) What is the magnitude of the indnced emf if the radins in part (e) is \(R ?(g)\) What is the induced emf if the radius in part \((e)\) is 2\(R ?\)

Terminal Speed. A bar of length \(L=0.8 \mathrm{m}\) is free to shide without friction on horizontal rails, as shown in Fig. 29.48 . There is a uniform magnetic ficld \(B=1.5 \mathrm{T}\) directed into the plane of the figure. At one end of the rails there is a battery with emf \(\mathcal{E}=12 \mathrm{V}\) and a switch. The bar has mass 0.90 \(\mathrm{kg}\) and resistance \(5.0 \Omega,\) and all other resistance in the circuit can be ignored. The switch is closed at time \(t=0\) . (a) Sketch the speed of the bar as a function of time. (b) Just aner the switch is closed, what is the acceleration of the bar? (c) What is the acceleration of the bar when its speed is 2.0 \(\mathrm{m} / \mathrm{s} ?\) (d) What is the terminal speed of the bar?

A metal ring 4.50 \(\mathrm{cm}\) in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic ficld. These magnets produce an initial uniform field of 1.12 T between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.250 \(\mathrm{T} / \mathrm{s}\) (a) What is the magnitude of the electric field induced in the ring? (b) In which direction (clockwise or counterclockwise) does the current flow as viewed by someone on the south pole of the magnet?
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