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Chapter 21: Problem 88

A 125-turn rectangular coil of wire is hung from one arm of a balance, as the figure shows. With the magnetic field \(\overrightarrow{\mathbf{B}}\) turned off, an object of mass \(M\) is added to the pan on the other arm to balance the weight of the coil. Concepts: (i) In a balanced, or equilibrium, condition the device has no angular acceleration. What does this imply about the net torque acting on the device? (ii) What is torque? (iii) In calculating the torques acting on an object in equilibrium, where do you locate the axis of rotation? Calculations: When a constant 0.200-T magnetic field is turned on and there is a current of \(8.50 \mathrm{A}\) in the coil, how much additional mass \(m\) must be added to regain the balance?

Short Answer

Expert verified
Calculate torque using magnetic force, equate to gravitational torque, solve for added mass m.

Step by step solution

01

Understand Equilibrium and Torque

When the device is in equilibrium, it means there's no net angular acceleration. This implies that the net torque acting on the system is zero. Torque itself is a measure of the rotational force acting on an object, calculated as the product of force and the distance from the axis of rotation.
02

Determine Axis of Rotation

For calculating torque, we often choose a convenient point as the axis of rotation. In this problem, let's consider the axis of rotation at the point where the coil is hung, balancing torques due to gravitational force and the magnetic force.
03

Calculate Magnetic Force on the Coil

The magnetic force acting on a coil can be calculated using the formula: \( F = n imes I imes L imes B \) where \( n = 125 \) is the number of turns, \( I = 8.50 \) A is the current, \( L \) is the length of the coil, and \( B = 0.200 \) T is the magnetic field. Since the coil forms a rectangular loop, force acts on the longer sides.
04

Calculate Torque Due to Magnetic Force

The magnetic torque can be calculated using the formula: \( \tau = n imes I imes A \times B \) where \( A \) is the area of the coil. To simplify, if given the dimension or if not calculate taking any standard length, then substitute back to find torque.
05

Balance the Torques

In equilibrium, the torque due to the magnetic force must be countered by the weight of additional mass. The gravitational torque can be equated as: \( au_{gravity} = m imes g \times d \) where \( g = 9.8 \) m/s虏 is gravitational acceleration and \( d \) is the distance from the axis, to solve for \( m \).
06

Calculate the Additional Mass

Equate the torque from the magnetic force to the gravitational torque to find the additional mass \( m \): \[ n imes I imes B \times L/2 = m imes g \times d/2 \] Using values \( g = 9.8 \) m/s虏 and simplifying to find \( m \).

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Key Concepts

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

Magnetic Force
Magnetic force is a fascinating interaction that occurs between magnetic fields and electric charges. In the context of our problem, it refers to the force exerted by the magnetic field on the wire coil. This force is calculated using the formula:
  • \( F = n \times I \times L \times B \)
where:
  • \(n\) is the number of turns in the coil,
  • \(I\) is the current flowing through the coil,
  • \(L\) is the length of the coil, and
  • \(B\) is the magnetic field's strength.
The magnetic force acts perpendicular to the flow of current and the magnetic field lines, causing the rectangular coil to experience a rotational push. This force is fundamental in causing a torque, which we need to balance with the gravitational torque to maintain equilibrium.
Angular Acceleration
Angular acceleration refers to the rate at which the angular velocity of an object changes with time. In simple terms, it's how quickly an object starts to rotate or stop rotating. In equilibrium, the net angular acceleration of a system is zero, meaning that any torques acting on it are canceling each other out. For instance, in our exercise, when the magnetic force is applied to the coil, the whole setup begins to rotate unless balanced. The absence of angular acceleration under equilibrium conditions means the magnetic and gravitational torques are perfectly balanced. This lack of movement allows the balance to maintain the steady state by perfectly adjusting the weights.
Axis of Rotation
The axis of rotation is a crucial concept when solving problems involving torque. It is an imaginary line that an object rotates around. Selecting the correct axis of rotation is key to accurately calculating the net torque, as torque depends on the distance from this axis to the line of action of the force. In the case of the wire coil hung on a balance, the logical axis of rotation is the point where the coil is suspended. This choice simplifies calculations because it allows us to focus on the torques due to both gravitational forces and magnetic forces. With equilibrium in mind, we ensure that the torques about this axis balance out. This approach yields a clearer understanding of how altering variables, like added mass or magnetic field strength, affects the system's rotational equilibrium.

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Most popular questions from this chapter

A charge of \(-8.3 \mu \mathrm{C}\) is traveling at a speed of \(7.4 \times 10^{6} \mathrm{m} / \mathrm{s}\) in a region of space where there is a magnetic field. The angle between the velocity of the charge and the field is \(52^{\circ} .\) A force of magnitude \(5.4 \times 10^{-3} \mathrm{N}\) acts on the charge. What is the magnitude of the magnetic field?

Electron beams are sometimes used to melt and evaporate metals in order to deposit thin metallic films on surfaces (similar to gold plating). One method is to put the material to be evaporated (called the "target") into a small tungsten cup (a crucible that has a very high melting point) and direct a beam of electrons at the target. Your team has been given the task of designing an electron-beam evaporator. The crucible is a cylinder, \(2.0 \mathrm{cm}\) in diameter and \(1.5 \mathrm{cm}\) in height, and contains a small target of pure nickel (Ni). The electrons are accelerated through a potential difference of \(V=1.20 \mathrm{kV}\), and form a beam that originates below the crucible, exactly \(3.70 \mathrm{cm}\) off its center, in the \(+x\) direction (see the drawing). (a) What is the speed of the electrons in the beam? (b) You must steer the electron beam with a magnetic field so that it curls over the lip of the cup and strikes the nickel target. Assuming that a uniform field exists above the cup (the field is zero below), what must be the radius of the beam's circular path? (c) In what direction should the field point if the beam initially approaches the cup from the \(-y\) axis? (d) What must be the magnitude of the uniform magnetic field?

A wire has a length of \(7.00 \times 10^{-2} \mathrm{m}\) and is used to make a circular coil of one turn. There is a current of \(4.30 \mathrm{A}\) in the wire. In the presence of a \(2.50-\mathrm{T}\) magnetic field, what is the maximum torque that this coil can experience?

When beryllium- 7 ions \(\left(m=11.65 \times 10^{-27} \mathrm{kg}\right)\) pass through a mass spectrometer, a uniform magnetic field of \(0.283 \mathrm{T}\) curves their path directly to the center of the detector (see Figure 21.14 ). For the same accelerating potential difference, what magnetic field should be used to send beryllium-10 ions \(\left(m=16.63 \times 10^{-27} \mathrm{kg}\right)\) to the same location in the detector? Both types of ions are singly ionized \((q=+e)\).

A very long, straight wire carries a current of 0.12 A. This wire is tangent to a single-turn, circular wire loop that also carries a current. The directions of the currents are such that the net magnetic field at the center of the loop is zero. Both wires are insulated and have diameters that can be neglected. How much current is there in the loop?
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