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Chapter 27: Problem 32

An electromagnet produces a magnetic field of \(0.550 \mathrm{~T}\) in a cylindrical region of radius \(2.50 \mathrm{~cm}\) between its poles. A straight wire carrying a current of 10.8 A passes through the center of this region and is perpendicular to both the axis of the cylindrical region and the magnetic field. What magnitude of force does this field exert on the wire?

Short Answer

Expert verified
The magnetic field exerts a force of \(5.94 \mathrm{~N/m}\) on the wire.

Step by step solution

01

Identify the given quantities

The magnetic field \(B\) is given as \(0.550 \mathrm{~T}\). The current \(I\) in the wire is given as \(10.8 \mathrm{~A}\). The direction of the current is perpendicular to the magnetic field, so \(\sin{\theta} = 1\).
02

Apply the formula for force on a current carrying wire in a magnetic field

Using the formula \(F = ILB \sin{\theta}\), and substituting the given information, we get: \(F = (10.8 \mathrm{~A}) \cdot L \cdot (0.550 \mathrm{~T}) \cdot 1\), where \(L\) is the length of the wire. Since the length isn't given and we're asked about the force, we interpret this formula as the force per unit length that the magnetic field exerts on the wire, therefore we rewrite the formula as: \(F/L = (10.8 \mathrm{~A}) \cdot (0.550 \mathrm{~T})\).
03

Calculate the force

We can now calculate the force per unit length that the magnetic field exerts on the wire: \(F/L = (10.8 \mathrm{~A}) \cdot (0.550 \mathrm{~T}) = 5.94 \mathrm{~N/m}\).

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Key Concepts

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

Electromagnetism
Electromagnetism is a branch of physics that focuses on the study of electric charges and their interactions through electric and magnetic fields. This fundamental theory explains how electric currents produce magnetic fields and how changing magnetic fields generate electric currents. It encompasses several phenomena:
  • Electric Fields: Produced by stationary electric charges.
  • Magnetic Fields: Produced by moving electric charges or currents.
  • Electromagnetic Induction: The process of generating electric currents from changing magnetic fields.
  • Maxwell's Equations: A set of mathematical equations describing how electric and magnetic fields interact.
In the given exercise, electromagnetism is at play through the interaction of the magnetic field with the current-carrying wire. This interaction is fundamental to many applications, including electric motors and transformers.
Magnetic Field
A magnetic field is a vector field surrounding magnets and electric currents. It exerts forces on other nearby moving charges or currents. The strength and direction of a magnetic field are represented by magnetic field lines. Key points to remember about magnetic fields include:
  • Magnetic Field Strength (B): Often measured in Tesla (T), this represents how strong the magnetic field is.
  • Magnetic Field Lines: They show the direction of the magnetic field. The closer these lines are, the stronger the magnetic field.
  • Magnetic Force: When a charged particle moves through a magnetic field, it experiences a force perpendicular to both its velocity and the field direction. This is calculated using F = qvB ightarrow, where q is charge and v is velocity.
In the problem, the magnetic field with a strength of 0.550 T surrounds the wire. This field, perpendicular to the wire, exerts a force on the current traveling through it.
Current-Carrying Wire
A current-carrying wire generates its magnetic field, creating interactions with external magnetic fields. When an external magnetic field is applied, it affects the wire depending on the current's direction:
  • Direction of Current: The direction of the current determines the direction of the magnetic field around the wire, following the right-hand rule.
  • Magnetic Force on Wire: The force exerted on a wire by a magnetic field is determined by the equation: \( F = ILB \sin\theta \), where I is the current, L is the length of the wire in the field, B is the magnetic field strength, and \( \sin\theta \) represents the angle between the current direction and the magnetic field. In the exercise, this angle is 90掳, making \( \sin\theta = 1 \).
  • Applications: Such interactions are used in many technologies, such as electric generators and loudspeakers.
Here, the wire carrying a current of 10.8 A is placed in a magnetic field of 0.550 T, resulting in a force calculated per unit length of 5.94 N/m.

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Most popular questions from this chapter

A flat circular coil carrying a current of 8.80 A has a magnetic dipole moment of \(0.194 \mathrm{~A} \cdot \mathrm{m}^{2}\) to the left. Its area vector \(A\) is \(4.0 \mathrm{~cm}^{2}\) to the left. (a) How many turns does the coil have? (b) An observer is on the coil's axis to the left of the coil and is looking toward the coil. Does the observer see a clockwise or counterclockwise current? (c) If a huge \(45.0 \mathrm{~T}\) external magnetic field directed out of the paper is applied to the coil, what torque (magnitude and direction) results?

A circular area with a radius of \(6.50 \mathrm{~cm}\) lies in the \(x y\) -plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field \(B=0.230 \mathrm{~T}\) (a) in the \(+z\) -direction; (b) at an angle of \(53.1^{\circ}\) from the \(+z\) -direction; (c) in the \(+y\) -direction?

Magnetic Moment of the Hydrogen Atom. In the Bohr model of the hydrogen atom (see Section 39.3 ), in the lowest energy state the electron orbits the proton at a speed of \(2.2 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in a circular orbit of radius \(5.3 \times 10^{-11} \mathrm{~m}\). (a) What is the orbital period of the electron? (b) If the orbiting electron is considered to be a current loop, what is the current \(I ?\) (c) What is the magnetic moment of the atom due to the motion of the electron?

You are using a type of mass spectrometer to measure charge-to-mass ratios of atomic ions. In the device, atoms are ionized with a beam of electrons to produce positive ions, which are then accelerated through a potential difference \(V\). (The final speed of the ions is great enough that you can ignore their initial speed.) The ions then enter a region in which a uniform magnetic field \(\vec{B}\) is perpendicular to the velocity of the ions and has magnitude \(B=0.250 \mathrm{~T}\). In this \(\overrightarrow{\boldsymbol{B}}\) region, the ions move in a semicircular path of radius \(R .\) You measure \(R\) as a function of the accelerating voltage \(V\) for one particular atomic ion: $$ \begin{array}{l|lllll} \boldsymbol{V}(\mathbf{k} \mathbf{V}) & 10.0 & 12.0 & 14.0 & 16.0 & 18.0 \\ \hline \boldsymbol{R}(\mathrm{cm}) & 19.9 & 21.8 & 23.6 & 25.2 & 26.8 \end{array} $$ (a) How can you plot the data points so that they will fall close to a straight line? Explain. (b) Construct the graph described in part (a). Use the slope of the best-fit straight line to calculate the charge-to-mass ratio \((q / m)\) for the ion. \((\mathrm{c})\) For \(V=20.0 \mathrm{kV},\) what is the speed of the ions as they enter the \(\vec{B}\) region? (d) If ions that have \(R=21.2 \mathrm{~cm}\) for \(V=12.0 \mathrm{kV}\) are singly ionized, what is \(R\) when \(V=12.0 \mathrm{kV}\) for ions that are doubly ionized?

A deuteron (the nucleus of an isotope of hydrogen) has a mass of \(3.34 \times 10^{-27} \mathrm{~kg}\) and a charge of \(+e .\) The deuteron travels in a circular path with a radius of \(6.96 \mathrm{~mm}\) in a magnetic field with magnitude \(2.50 \mathrm{~T}\). (a) Find the speed of the deuteron. (b) Find the time required for it to make half a revolution. (c) Through what potential difference would the deuteron have to be accelerated to acquire this speed?
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