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Chapter 29: Problem 4

A closely wound search coil (Exercise 29.3) has an area of \(3.20 \mathrm{cm}^{2}, 120\) turns, and a resistance of \(60.0 \Omega .\) It is connected to a charge-measuring instrument whose resistance is 45.0\(\Omega\) . When the coil is rotated quickly from a position parallel to a uniform magnetic field to a position perpendicular to the field, the instrument indicates a charge of \(3.56 \times 10^{-5} \mathrm{C}\) . What is the magnitude of the field?

Short Answer

Expert verified
The magnetic field magnitude is 0.0974 T.

Step by step solution

01

Convert Units

First, convert the area of the coil from square centimeters to square meters. Recall that 1 cm extsuperscript{2} = 10 extsuperscript{-4} m extsuperscript{2}. Therefore, the area in square meters is \( 3.20 \times 10^{-4} \text{ m}^2 \).
02

Calculate Total Resistance

The total resistance in the circuit is the sum of the resistance of the coil and the charge-measuring instrument. Thus, the total resistance \( R_{\text{total}} = 60.0 \Omega + 45.0 \Omega = 105.0 \Omega \).
03

Apply Faraday's Law of Induction

Faraday's Law states that the induced electromotive force (emf) in the coil \( \mathcal{E} \) can be calculated by the formula \( \mathcal{E} = -N \frac{d\Phi}{dt} \), where \( N \) is the number of turns and \( \Phi \) is the magnetic flux. The change in magnetic flux \( \Delta \Phi = B \cdot A \cdot (\cos(90^\circ) - \cos(0^\circ)) = -B \cdot A \).
04

Find the Induced EMF

The total charge \( Q \) measured is related to the induced EMF by \( Q = \mathcal{E} \cdot \frac{R}{R_{\text{total}}} \). Rearrange for \( \mathcal{E} \): \( \mathcal{E} = Q \times R_{\text{total}} \). Given \( Q = 3.56 \times 10^{-5} \mathrm{C} \), \( \mathcal{E} = 3.56 \times 10^{-5} \mathrm{C} \times 105.0 \Omega = 3.738 \times 10^{-3} \mathrm{V} \).
05

Calculate the Magnetic Field Magnitude

Substitute the expression for \( \mathcal{E} \) into the expression for flux change: \( \mathcal{E} = N \cdot \Delta \Phi / \Delta t = N \cdot B \cdot A \). Solve for \( B \) by rearranging: \( B = \frac{\mathcal{E}}{N \cdot A} \). Substitute the known values: \( B = \frac{3.738 \times 10^{-3} \mathrm{V}}{120 \times 3.20 \times 10^{-4} \text{ m}^2} = 0.0974 \mathrm{T} \).

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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 key concept in electromagnetism, often symbolized as \( \Phi \). It represents the quantity of magnetic field lines passing through a given area. Imagine magnetic flux as the number of field lines that penetrate the surface of an object, such as a coil. This value helps us to understand how changes in the magnetic field affect electromagnetic phenomena.

To calculate magnetic flux, use the formula \( \Phi = B \cdot A \cdot \cos(\theta) \), where:
  • \( B \) is the magnetic field strength (in teslas).
  • \( A \) is the area through which the field lines pass (in square meters).
  • \( \theta \) is the angle between the magnetic field lines and the perpendicular to the surface.
When the plane of the coil is rotated, the angle \( \theta \) changes, affecting the magnetic flux and inducing an electromotive force.
Induced Electromotive Force (EMF)
The induced electromotive force, often abbreviated as EMF, is an essential concept in Faraday's Law of Induction. It denotes the voltage generated in a coil due to the change in magnetic flux over time. According to Faraday's Law, the induced EMF \( \mathcal{E} \) is given by the relationship \( \mathcal{E} = -N \frac{d\Phi}{dt} \), where:
  • \( N \) is the number of turns in the coil.
  • \( \frac{d\Phi}{dt} \) represents the rate of change of magnetic flux.
This negative sign follows Lenz's Law, indicating that the induced EMF opposes the change in magnetic flux. When a coil is rotated in a magnetic field, the changing orientation causes a variation in flux, thus inducing an EMF. This induced voltage is key in many applications, such as generators and transformers.
Circuit Resistance
Circuit resistance is a measurement of how much a circuit opposes the flow of electric current, expressed in ohms (\( \Omega \)). In a practical scenario, the total resistance \( R_{\text{total}} \) of a system is the sum of resistances in the different components connected in series. For example, in the search coil circuit, the total resistance is due to both the coil's resistance and the resistance of the measurement instrument.

Calculating the total resistance helps in determining the overall behavior of the circuit. It affects how the induced current and EMF behave, especially when the coil undergoes changes in magnetic environments. Remember, higher resistance results in less current for a given applied voltage.
Coil Turns
Coil turns refer to the number of windings or loops in a coil. Each loop contributes to the total electromotive force when exposed to a changing magnetic field, due to the collective effect on the magnetic flux linkage. The number of turns, denoted as \( N \), amplifies the induced EMF, as evidenced by the formula \( \mathcal{E} = N \cdot \frac{d\Phi}{dt} \).

In practical applications, a high number of turns in a coil increases the sensitivity and effectiveness of devices that rely on electromagnetic induction. This attribute is crucial in equipment such as transformers, inductors, or search coils, where the induction of voltage is desired. Therefore, understanding and manipulating the number of coil turns is vital for optimizing electromagnetic performance.

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Most popular questions from this chapter

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.

A circular loop of wire with a radius of 12.0 \(\mathrm{cm}\) and oriented in the horizontal \(x y\) -plane is located in a region of uniform magnetic field. A field of 1.5 \(\mathrm{T}\) is directed along the positive z-direction, which is upward. (a) If the loop is removed from the field region in a time interval of 2.0 \(\mathrm{ms}\) , find the average emf that will be induced in the wire loop during the extraction process. (b) If the coil is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

emf in a Bullet. At the equator, the earth's magnetic field is approximately horizontal, is directed towand the north, and has a value of \(8 \times 10^{-5} \mathrm{T}\) . (a) Estimate the emf induced between the top and bottom of a bullet shot horizontally at a target on the equator if the bullet is shot toward the east. Assume the bullet has a length of 1 \(\mathrm{cm}\) and a diamcter of 0.4 \(\mathrm{cm}\) and is traveling at 300 \(\mathrm{m} / \mathrm{s}\) . Which is at higher potential: the top or boutom of the bullet? (b) What is the emfif the bullet travels south?(c) What is the emf induced between the front and back of the bullet for any horizontal velocity?

An airplane propeller of total length \(L\) rotates around its center with angular spced \(\omega\) in a magnctic ficld that is perpcndicular to the plane of rotation. Modeling the propeller as a thin, uniform bar, find the potential difference between (a) the center and either end of the propeller and (b) the two ends. (c) If the field is the earth's field of 0.50 \(\mathrm{G}\) and the propeller turns at 220 \(\mathrm{rpm}\) and is 2.0 \(\mathrm{m}\) long, what is the potential difference between the middle and either end? It this large enough to be concemed about?

Falling Square Loop. A vertically oricnted, square loop of copper wire falls from a region where the field \(\overrightarrow{\boldsymbol{B}}\) is horizontal. uniform. where the field is zero. The loop is released from rest and initially is entirely within the magnetic-field region. Let the side length of the loop be \(s\) and let the diameter of the wire be \(d\) . The resistivity of copper is \(\rho_{R}\) and the density of copper is \(\rho_{m}\) . If the loop reaches its terminal speed while its upper segment is still in the magnetic- field region, find an expression for the terminal speed.
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