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Chapter 19: Problem 45

A straight wire segment of length \(25 \mathrm{cm}\) carries a current of $33.0 \mathrm{A}$ and is immersed in a uniform external magnetic field. The magnetic force on the wire segment has magnitude \(4.12 \mathrm{N} .\) (a) What is the minimum possible magnitude of the magnetic field? (b) Explain why the given information enables you to calculate only the minimum possible field strength.

Short Answer

Expert verified
Answer: The minimum possible magnetic field strength is 0.0499 T.

Step by step solution

01

Recall the formula for magnetic force

The formula for magnetic force on a current-carrying wire in a magnetic field is given by: \(F = BIL\sin\theta\), where \(F\) is the magnetic force, \(B\) is the magnetic field strength, \(I\) is the current, \(L\) is the length of the wire, and \(\theta\) is the angle between the direction of the current and the magnetic field.
02

Identify the given values and what needs to be found

We are given the values for the length of the wire \(L = 25\,\text{cm} = 0.25\,\text{m}\), the current \(I = 33.0\,\text{A}\), and the force \(F = 4.12\,\text{N}\). Our goal is to find the minimum possible magnetic field strength \(B_{\text{min}}\).
03

Find the condition for minimum magnetic field strength

The magnetic field strength will be minimum when the angle \(\theta\) between the direction of the current and the magnetic field is \(90^{\circ}\), as the sine function becomes maximum (\(\sin90^{\circ} = 1\)). In this case, \(F = B_{\text{min}}IL\).
04

Calculate the minimum magnetic field strength

Using the formula mentioned in the previous step, we can find \(B_{\text{min}}\): \(B_{\text{min}} = \frac{F}{IL} = \frac{4.12\,\text{N}}{(33.0\,\text{A})(0.25\,\text{m})} = 0.0499\,\text{T}\)
05

Explanation for why the given information enables to calculate only the minimum possible field strength

The minimum possible field strength is calculated based on the condition that the angle \(\theta\) between the current and the magnetic field is \(90^{\circ}\), which maximizes the force acting on the wire. However, we do not have information about the actual angle \(\theta\) in the problem, and therefore cannot calculate a specific field strength if \(\theta\) is not \(90^{\circ}\). The problem only provides enough information to find the minimum possible field strength, which is when the force is maximized and the sine component is at its maximum value (1).

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Most popular questions from this chapter

An electromagnetic rail gun can fire a projectile using a magnetic field and an electric current. Consider two conducting rails that are \(0.500 \mathrm{m}\) apart with a \(50.0-\mathrm{g}\) conducting rod connecting the two rails as in the figure with Problem \(46 .\) A magnetic field of magnitude \(0.750 \mathrm{T}\) is directed perpendicular to the plane of the rails and rod. A current of \(2.00 \mathrm{A}\) passes through the rod. (a) What direction is the force on the rod? (b) If there is no friction between the rails and the rod, how fast is the rod moving after it has traveled \(8.00 \mathrm{m}\) down the rails?

Imagine a long straight wire perpendicular to the page and carrying a current \(I\) into the page. Sketch some \(\overrightarrow{\mathbf{B}}\) field lines with arrowheads to indicate directions.
Electrons in a television's CRT are accelerated from rest by an electric field through a potential difference of \(2.5 \mathrm{kV} .\) In contrast to an oscilloscope, where the electron beam is deflected by an electric ficld, the beam is deflected by a magnetic field. (a) What is the specd of the electrons? (b) The beam is deflected by a perpendicular magnetic field of magnitude $0.80 \mathrm{T}$. What is the magnitude of the acceleration of the electrons while in the field? (c) What is the speed of the electrons after they travel 4.0 mm through the magnetic field? (d) What strength electric field would give the electrons the same magnitude acceleration as in (b)? (c) Why do we have to use an clectric ficld in the first place to get the electrons up to speed? Why not use the large acceleration due to a magnetic field for that purpose?
In an electric motor, a circular coil with 100 turns of radius $2.0 \mathrm{~cm}$ can rotate between the poles of a magnet. When the current through the coil is \(75 \mathrm{~mA},\) the maximum torque that the motor can deliver is \(0.0020 \mathrm{~N} \cdot \mathrm{m} .\) (a) What is the strength of the magnetic field? (b) Is the torque on the coil clockwise or counterclockwise as viewed from the front at the instant shown in the figure?
A straight wire segment of length \(0.60 \mathrm{m}\) carries a current of $18.0 \mathrm{A}$ and is immersed in a uniform external magnetic ficld of magnitude \(0.20 \mathrm{T}\). (a) What is the magnitude of the maximum possible magnetic force on the wire segment? (b) Explain why the given information enables you to calculate only the maximum possible force.
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