/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 A loudspeaker creates sound by p... [FREE SOLUTION] | 91Ó°ÊÓ

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A loudspeaker creates sound by pushing air back and forth with a paper cone that is driven by a magnetic force on a wire coil at the base of the cone. Figure \(\mathrm{P} 24.52\) shows the details. The cone is attached to a coil of wire that sits in the gap between the poles of a circular magnet. The 0.18 T magnetic field, which points radially outward from \(\mathrm{N}\) to \(\mathrm{S}\), exerts a force on a current in the wire,moving the cone. The coil of wire that sits in this gap has a diameter of \(5.0 \mathrm{cm},\) contains 20 turns of wire, and has a resistance of \(8.0 \Omega .\) The speaker is connected to an amplifier whose instantaneous output voltage of \(6.0 \mathrm{V}\) creates a clockwise current in the coil as seen from above. What is the magnetic force on the coil at this instant?

Short Answer

Expert verified
The magnetic force on the coil at this instant is 0.42 N

Step by step solution

01

Calculate the Current

Ohm's law states that the current \(I\) in a circuit is equal to the voltage \(V\) divided by the resistance \(R\). Using this, the current \(I\) in the coil can be calculated as \(I = V / R\). Substituting the given voltage of 6.0 V and the resistance of 8.0 Ω into the formula gives \(I = 6.0 \, \text{V} / 8.0 \, \Omega = 0.75 \, \text{A}\).
02

Calculate the Magnetic Force

The force \(F\) on a current \(I\) in a magnetic field with magnetic field strength \(B\) and length \(l\) is given by the equation \(F = B I l \sin(\theta)\), where \(\theta\) is the angle between the direction of the current and the magnetic field. The angle in this case is 90 degrees because the magnetic field is radially outward and the current is clockwise, so \(\sin(90)\) equals 1. The length of wire in the magnetic field can be calculated by multiplying the total number of turns of the coil by the circumference of the coil, \(l = 2 \pi r N\), where \(r\) is the radius of the coil, \(N\) is the number of turns. Plugging in the given diameter of 5.0 cm (therefore radius of 2.5 cm or 0.025 m), the number of turns of 20, we get \(l = 2 \pi 0.025 \, \text{m} 20 = 3.14 \, \text{m}\). Therefore, the force \(F\) equals \(F = 0.18 \, \text{T} 0.75 \, \text{A} 3.14 \, \text{m} \sin(90) = 0.42 \, \text{N}\).
03

Answer the Question

With the calculations made, it is deduced that the magnetic force on the coil at this instant is 0.42 N.

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Key Concepts

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

Ohm's Law
Ohm's Law is a fundamental principle in the field of electricity and electronics, expressing the relationship between voltage (V), current (I), and resistance (R). It states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points. Formally, Ohm's Law is given by the equation:
\[ I = \frac{V}{R} \]
This formula implies that if you increase the voltage, the current will increase, provided the resistance stays constant. Conversely, if resistance increases, with a constant voltage, the current will decrease. Understanding this principle is crucial when dealing with electrical circuits, like in our example with the loudspeaker where the current in the coil was calculated using this law.
To visualize this, imagine pushing water through a pipe. Ohm’s Law is like saying the amount of water that flows (current) depends on the water pressure (voltage) and the size of the pipe (resistance). In the context of the exercise, knowing the voltage supplied by the amplifier and the coil's resistance, we could easily calculate the current that influences the magnetic force on the coil.
Magnetic Field Strength
Magnetic field strength, often denoted by the symbol 'B', quantifies the magnitude of a magnetic field at any given point in space, measured in teslas (T) in the SI system. It is a vector quantity, which means it has both a magnitude and a direction. The strength of the magnetic field is a key determinant of the magnetic force exerted on a moving charge or current-carrying conductor.
Consider the magnetic field strength as the 'muscle' of a magnet; the stronger this 'muscle', the more significant the effect it can have on magnetic objects within its reach. In our textbook example with the loudspeaker, the magnetic field strength of 0.18 T is inherently related to the force acting on the coil. The direction of the magnetic field, radially outward in this case, also impacts the direction of the force. As the magnetic field interacts with the electric current flowing through the coil, it generates a force acting perpendicular to both the current's direction and the magnetic field lines, ultimately causing the movement of the speaker cone.
Current in a Wire
  • Current is the flow of electric charge carriers, such as electrons.
  • The unit of current is the ampere (A).
  • A higher current implies a larger number of electrons moving through the wire per unit time.
In a wire, current is like the flow of water through a pipe, and the wire can be thought of as a channel that guides the flow of electricity. The amount of current has direct implications for the magnetic force experienced by a wire in a magnetic field. Based on the right-hand rule, if you point your thumb in the direction of the current and your fingers in the direction of the magnetic field, your palm indicates the direction of the force on the wire.
In the loudspeaker scenario, the current generated in the coil, due to the applied voltage from the amplifier, interacts with the magnetic field of the circular magnet. This interaction between the current and magnetic field results in the force that drives the cone of the loudspeaker to create sound waves. Our example illustrates how the current flowing in a specific direction - clockwise as viewed from above - plays a critical role in how the cone moves.

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Most popular questions from this chapter

In a simplified model of the hydrogen atom, its electron moves in a circular orbit with a radius of \(5.3 \times 10^{-10} \mathrm{m}\) at a frequency of \(6.6 \times 10^{15}\) Hz. What magnetic field would be required to cause an electron to undergo this same motion?

Young domestic chickens have the ability to orient themselves in the earth's magnetic field. Researchers used a set of two coils to adjust the magnetic field in the chicks' pen. Figure P24.20 shows the two coils, whose centers coincide, seen edge-on. The axis of coil 1 is parallel to the ground and points to the north; the axis of coil 2 is oriented vertically. Each coil has 43 turns and a radius of \(1.0 \mathrm{m}\). At the location of the experiment, the earth's field had a magnitude of \(5.6 \times 10^{-5} \mathrm{T}\) and pointed to the north, tilted up from the horizontal by \(61^{\circ} .\) If the researchers wished to exactly cancel the earth's field, what currents would they need to apply in coil 1 and in coil \(2 ?\)

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A long wire carrying a \(5.0 \mathrm{A}\) current perpendicular to the \(x y\) -plane intersects the \(x\) -axis at \(x=-2.0 \mathrm{cm} .\) A second, parallel wire carrying a \(3.0 \mathrm{A}\) current intersects the \(x\) -axis at \(x=+2.0 \mathrm{cm} .\) At what point or points on the \(x\) -axis is the magnetic field zero if (a) the two currents are in the same direction and (b) the two currents are in opposite directions?

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