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Chapter 28: Problem 82

To use a larger sample, the experimenters construct a solenoid that has the same length, type of wire, and loop spacing but twice the diameter of the original. How does the maximum possible magnetic torque on a bacterium in this new solenoid compare with the torque the bacterium would have experienced in the original solenoid? Assume that the currents in the solenoids are the same. The maximum torque in the new solenoid is (a) twice that in the original one; (b) half that in the original one; (c) the same as that in the original one; (d) one-quarter that in the original one.

Short Answer

Expert verified
The maximum torque in the new solenoid is (c) the same as that in the original one.

Step by step solution

01

Understanding the context

The magnetic field \(B\) inside an ideal solenoid is uniform and parallel to the axis of the solenoid. It is represented by the formula \(B = 渭_n I\) where \(渭\) is the magnetic constant, \(n\) is the number of loops per unit length and \(I\) is the current flowing through the solenoid. Torque \(\tau\) experienced by a magnetic dipole moment \(渭\) in a uniform magnetic field \(B\) is given by \(\tau = 渭Bsin胃\), where \(胃\) is the angle between the direction of \(B\) and \(渭\).
02

Analyzing the change

From step 1, we can see that the parameters that have been changed in the problem (the diameter of the solenoid) do not affect the magnetic field \(B\) and hence the maximum magnetic torque \(\tau\) on the bacterium would not change.
03

Verifying the result

The number of loops per unit length, the current, and the length of the solenoid remain the same in the new solenoid, therefore, the magnetic field \(B\) and the magnetic torque \(\tau\) remain the same as well.

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Key Concepts

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

Solenoid
A solenoid is a coil of wire often used to generate a magnetic field when an electric current is passed through it. The wire is wound in a helix, usually in the shape of a cylinder. Solenoids are especially vital in electromagnetism, as they can create a uniform magnetic field in their interior, making them useful in applications like electromagnets, inductors, and transformers.

When a solenoid is long and closely wound, the magnetic field inside is relatively uniform. This uniformity is beneficial for experiments requiring consistent magnetic conditions. The magnetic field inside an **ideal solenoid** can be understood by the formula:
  • \( B = 渭_0 n I \)
where:
  • \( 渭_0 \) is the permeability of free space, a constant.
  • \( n \) is the number of turns per unit length of the solenoid.
  • \( I \) is the current running through the solenoid.
The diameter of a solenoid does not affect the magnetic field inside, so even if a solenoid's diameter is increased, its internal field remains unchanged as long as the other factors remain constant.
Magnetic Field
A magnetic field is a vector field that describes the magnetic influence of electrical currents and magnetized materials. The strength and direction of a magnetic field are denoted by the symbol \( B \), and it represents the ability to exert a force on moving charges and current-carrying wires.

In solenoids, the magnetic field is particularly crucial. Inside an ideal solenoid, the field is consistent and parallel to its axis:

  • The field inside ideally does not depend on the solenoid's diameter.
  • Instead, it depends linearly on the number of loops per unit length \( n \) and the current \( I \).
This uniformity allows solenoids to simulate strong magnetic environments needed in experiments and applications like Magnetic Resonance Imaging (MRI) machines and various industrial machines where controlled magnetic conditions are necessary.
Magnetic Dipole Moment
The magnetic dipole moment is a measure of the strength and orientation of a magnet's ability to align with a magnetic field. It is represented by the symbol \( 渭_m \), and describes how a magnetic field interacts with a circuit or a molecule.

In a practical sense, it explains the torque a magnetic field exerts on a magnet or a magnetic body. **This torque** \( \tau \) can be calculated using the equation:
  • \( \tau = 渭_m B \sin 胃 \)
where:
  • \( B \) is the magnitude of the magnetic field.
  • \( \sin 胃 \) is the sine of the angle between the magnetic field and the dipole moment.
In the context of the exercise, the torque experienced by a bacterium with a magnetic dipole moment in a solenoid does not change with the diameter of the solenoid, because other factors like the magnetic field strength remain constant. This concept illustrates how magnetic properties are interrelated and why experimental conditions must be carefully controlled.

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Most popular questions from this chapter

A \(+6.00 \mu\) C point charge is moving at a constant \(8.00 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in the \(+y\) -direction, relative to a reference frame. At the instant when the point charge is at the origin of this reference frame, what is the magnetic-field vector \(\overrightarrow{\boldsymbol{B}}\) it produces at the following points: (a) \(x=0.500 \mathrm{~m}, y=0, z=0\) (b) \(x=0, y=-0.500 \mathrm{~m}, z=0\) (c) \(x=0, \quad y=0, \quad z=+0.500 \mathrm{~m}\) (d) \(x=0, \quad y=-0.500 \mathrm{~m}\) \(z=+0.500 \mathrm{~m} ?\)

A magnetic field of \(37.2 \mathrm{~T}\) has been achieved at the MIT Francis Bitter Magnet Laboratory. Find the current needed to achieve such a field (a) \(2.00 \mathrm{~cm}\) from a long, straight wire; (b) at the center of a circular coil of radius \(42.0 \mathrm{~cm}\) that has 100 turns; (c) near the center of a solenoid with radius \(2.40 \mathrm{~cm},\) length \(32.0 \mathrm{~cm},\) and 40,000 turns.

A wire of length \(20.0 \mathrm{~cm}\) lies along the \(x\) -axis with the center of the wire at the origin. The wire carries current \(I=8.00 \mathrm{~A}\) in the \(-x\) -direction. (a) What is the magnitude \(B\) of the magnetic field of the wire at the point \(y=5.00 \mathrm{~cm}\) on the \(y\) -axis? (b) What is the percent difference between the answer in (a) and the value you obtain if you assume the wire is infinitely long and use Eq. ( 28.9 ) to calculate \(B\) ?

A toroidal solenoid with 400 turns of wire and a mean radius of \(6.0 \mathrm{~cm}\) carries a current of 0.25 A. The relative permeability of the core is \(80 .\) (a) What is the magnetic field in the core? (b) What part of the magnetic field is due to the magnetic moments of the atoms in the core?

A closed curve encircles several conductors. The line integral \(\oint \overrightarrow{\boldsymbol{B}} \cdot d \overrightarrow{\boldsymbol{\imath}}\) around this curve is \(3.83 \times 10^{-4} \mathrm{~T} \cdot \mathrm{m} .\) (a) What is the net cur- rent in the conductors? (b) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral? Explain.
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