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Chapter 29: Problem 2

In a physics laboratory experiment, a coil with 200 tums enclosing an area of 12 \(\mathrm{cm}^{2}\) is rotated in 0.040 s from a position where its plane is perpendicular to the earth's magnetic field to a position where its plane is parallel to the field. The earth's magnetic field at the lab location is \(6.0 \times 0^{-5} \mathrm{T}\) . (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? b) What is the average emf induced in the coil?

Short Answer

Expert verified
a) Initial flux: \(7.2 \times 10^{-7} \text{Tm}^2\); final flux: 0. b) Average emf: \(3.6 \times 10^{-3} \text{V}\).

Step by step solution

01

Calculate Initial and Final Magnetic Flux

To calculate the magnetic flux, we need to use the formula \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( \Phi \) is the magnetic flux, \( B \) is the magnetic field strength, \( A \) is the area, and \( \theta \) is the angle between the magnetic field and the normal to the coil's plane. Initially, \( \theta = 0^{\circ} \), so \( \cos(\theta) = 1 \). The initial flux is \( \Phi_i = B \cdot A = 6.0 \times 10^{-5} \times 12 \times 10^{-4} = 7.2 \times 10^{-7} \text{Tesla meters squared (Tm}^2\text{)} \). After rotation, the plane is parallel to the field, \( \theta = 90^{\circ} \), so \( \cos(\theta) = 0 \). The final flux is \( \Phi_f = 0 \).
02

Calculate Change in Magnetic Flux

The change in magnetic flux \( \Delta \Phi \) is the difference between the initial and final flux. Thus, \( \Delta \Phi = \Phi_f - \Phi_i = 0 - 7.2 \times 10^{-7} = -7.2 \times 10^{-7} \text{Tm}^2 \).
03

Calculate Average Induced EMF

Using Faraday's law of electromagnetic induction, the average induced emf \( \mathcal{E} \) is given by \( \mathcal{E} = -N \frac{\Delta \Phi}{\Delta t} \), where \( N \) is the number of turns (200) and \( \Delta t \) is the time interval in seconds (0.040 s). Substitute the values: \( \mathcal{E} = -200 \times \frac{-7.2 \times 10^{-7}}{0.040} = 3.6 \times 10^{-3} \text{V} \). The negative sign from Faraday鈥檚 law indicates the direction of the induced emf, but we are interested in the magnitude here.

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Key Concepts

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

Magnetic Flux
Magnetic flux is a core concept in understanding electromagnetic induction. It represents the total magnetic field passing through a given area. Think of it like the number of magnetic field lines penetrating through the surface of a coil. This can be visualized as a density map for magnetic lines in a particular area.

The formula to calculate magnetic flux, \( \Phi \), is given as:
  • \( \Phi = B \cdot A \cdot \cos(\theta) \)
Here,
  • \( B \) is the magnetic field strength.
  • \( A \) is the area through which the field lines pass.
  • \( \theta \) is the angle between the magnetic field and the normal to the surface.
In this exercise, calculating the initial magnetic flux involved having the coil perpendicular to the magnetic field. This makes \( \cos(\theta) = 1 \), meaning the maximum number of magnetic field lines pass through.

After rotation, when the coil is parallel to the field, \( \theta = 90^\circ \), leading to \( \cos(\theta) = 0 \), signifying no lines pass through and thus zero magnetic flux. Understanding this relationship helps in grasping how changes in physical alignment can affect magnetic flux values.
Electromagnetic Induction
Electromagnetic induction is the process of generating an electromotive force (EMF) across a conductor when it is exposed to a changing magnetic field. Discovered by Michael Faraday, this phenomenon is foundational in the development of devices like transformers and generators.

Faraday's Law of Electromagnetic Induction states that the induced EMF in any closed circuit is equal to the negative rate of change of the magnetic flux through the circuit.
  • This can be mathematically represented by the formula: \( \mathcal{E} = - N \frac{\Delta \Phi}{\Delta t} \)
In simple terms, when the magnetic environment of a coil changes, either by moving the coil in a magnetic field or altering the field鈥檚 strength or direction, an EMF is induced. The minus sign indicates Lenz鈥檚 Law, specifying that the induced EMF will generate a current opposing the change in flux.

Understanding these principles is crucial in analyzing circuits and designing electronic systems that rely on magnetic force.
Coil in Magnetic Field
A coil positioned in a magnetic field experiences certain effects based on its orientation relative to the field. In the experiment, the coil started perpendicular to the earth's magnetic field before rotating to become parallel. This change in position is paramount to understanding how the experiment impacts the flux and the consequent EMF values.

In general:
  • A perpendicular coil position allows maximum magnetic field lines to pass through, maximizing flux.
  • A parallel position results in no lines passing through, minimizing flux.
Thus, changing the coil鈥檚 position relative to the field alters the flux. This change in flux is critical in applications where the movement of the coil or changes in the magnetic environment are utilized to generate electricity, informing the design of sensors and inking mechanisms in devices like microphones and electric guitars.
Induced EMF
The concept of induced EMF is central to Faraday鈥檚 experiments and to this problem. It refers to the voltage generated when a coil is exposed to a changing magnetic flux. This change can be caused by moving either the magnetic field or the coil itself.

To compute the average induced EMF, Faraday's Law provides:
  • \( \mathcal{E} = - N \frac{\Delta \Phi}{\Delta t} \)
In our exercise, \( N \) represents the number of turns in the coil, \( \Delta \Phi \) is the change in magnetic flux, and \( \Delta t \) is the time over which this change occurs. The calculated value, \( 3.6 \times 10^{-3} \) V, represents the EMF's magnitude generated by the coil鈥檚 rotation.

The negative sign in the formula is significant, indicating that the induced EMF's direction opposes the change in magnetic flux鈥攁n embodiment of Lenz's law. Thus, understanding the dynamics of induced EMF opens doors to harnessing electrical energy conversion systems, pivotal in engineering fields.

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Most popular questions from this chapter

It is impossible to have a uniform electric field that abruptly drops to zero in a region of space in which the magnetic field is constant and in which there are no electric charges. To prove this statement, use the method of contradiction: Assume that such a case is possible and then show that your assumption contradicts a law of nature. (a) In the bottom half of a piece of paper, draw evenly spaced horizontal lines representing a uniform electric field to your right. Use dashed lines to draw a rectangle abcda with horizontal side ab in the electric-field region and horizontal side \(c d\) in the top half of your paper where \(E=0 .\) (b) Show that integration around your rectangle contradicts Faraday's law, Eq. \((29.21) .\)

A \(1.41-\mathrm{m}\) bar moves through a uniform, 1.20 . \(T\) magnetic field with a speed of 2.50 \(\mathrm{m} / \mathrm{s}\) (Fig, 29.40\()\) . In cach case, find the emf induced between the ends of this bar and identify which, if any, end \((a \text { or } b)\) is at the higher potential. The bar moves in the direction of (a) the \(+x\) -axis; (b) the \(-y\) -axis; (c) the \(+z\) -axis. (d) How should this bar move so that the emf across its ends has the greatest possible value with \(b\) at a higher potential than \(a\) , and what is this maximum emf?

A capacitor has two parallel plates with area \(A\) separated by a distance \(d\) . The space between plates is filled with a material having dielectric constant \(K\) . The material is not a perfect insulator but has resistivity \(\rho\) . The capacitor is initially charged with charge of magnitude \(Q_{0}\) on each plate that gradually discharges by conduction through the dielectric. (a) Calculate the conduction current density \(j_{\mathrm{C}}(t)\) in the dielectric. (b) Show that at any instant the dis-placement current density in the diclectric is equal in magnitude to the oonduotion current density but opposite in direction, so the total current density is zero at every instant.

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\(R ?\)

A slender rod, 0.240 \(\mathrm{m}\) long, reates with an angular speed of 8.80 \(\mathrm{rad} / \mathrm{s}\) about an axis through one end and perpendicular to the rod. The plane of rotation of the rod is perpendicular to a uniform magnetic field with a magnitude of 0.650 \(\mathrm{T}\) (a) What is the induced emf in the rod? (b) What is the potential difference between its ends? (c) Suppose instead the rod rotates at 8.80 \(\mathrm{rad} / \mathrm{s}\) about an axis through its center and perpendicular to the rod, In this case, what is the potential difference between the ends of the rod? Between the center of the rod and one end?
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