/*! 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 13 A standard door into a house rot... [FREE SOLUTION] | 91影视

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Chapter 22: Problem 13

A standard door into a house rotates about a vertical axis through one side, as defined by the door's hinges. A uniform magnetic field is parallel to the ground and perpendicular to this axis. Through what angle must the door rotate so that the magnetic flux that passes through it decreases from its maximum value to one-third of its maximum value?

Short Answer

Expert verified
The door must rotate approximately 70.5掳.

Step by step solution

01

Understanding Magnetic Flux

Magnetic flux (\(\Phi\)) through a surface is given by the formula \(\Phi = B \cdot A \cdot \cos(\theta)\), where \(B\) is the magnetic field strength, \(A\) is the area of the surface, and \(\theta\) is the angle between the magnetic field direction and the normal (perpendicular) to the surface of the door.
02

Define Maximum Magnetic Flux

The maximum magnetic flux occurs when \(\theta = 0\), meaning the magnetic field is directly aligned with the normal to the surface of the door. In this case, \(\cos(0) = 1\), so the maximum flux is \(\Phi_{\text{max}} = B \cdot A\).
03

Express Given Condition

The exercise specifies that the flux should decrease to one-third of its maximum value. Thus, we want the flux \(\Phi\) to be given by: \(\frac{1}{3} \cdot \Phi_{\text{max}} = B \cdot A \cdot \cos(\theta)\).
04

Set up the Equation

Substitute \(\Phi_{\text{max}}\) from Step 2 into the equation from Step 3: \(\frac{1}{3} \cdot B \cdot A = B \cdot A \cdot \cos(\theta)\). Simplifying, we have \(\cos(\theta) = \frac{1}{3}\).
05

Solve for \(\theta\)

We need to find angle \(\theta\) such that \(\cos(\theta) = \frac{1}{3}\). Use the inverse cosine function (arccos) to find \(\theta\): \(\theta = \cos^{-1}\left(\frac{1}{3}\right)\).
06

Calculate the Angle

Using a calculator, compute \(\theta = \cos^{-1}\left(\frac{1}{3}\right)\approx 70.5^\circ\).

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Key Concepts

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

Magnetic Fields
Magnetic fields are invisible forces that can attract or repel certain materials. Imagine them as invisible lines of force that surround magnets and electric currents. These fields exert forces on moving charges, like electrons, and influence how they move. In the context of the exercise, the magnetic field is uniform, meaning its strength and direction are consistent in the region of interest, around the door. This uniformity makes it easier to calculate magnetic flux because the field's properties do not change over the surface area of interest. The magnetic field in this scenario is parallel to the ground and perpendicular to the door鈥檚 axis of rotation. This means that as the door rotates, the angle between the magnetic field and the normal to the door鈥檚 surface changes, affecting the magnetic flux. Understanding these characteristics of magnetic fields is crucial in predicting how they interact with surfaces like the door.
Angle of Rotation
The angle of rotation in this problem is key to determining how much the magnetic flux changes as the door swings. When the door is fully open or closed, it aligns with the maximum or minimum magnetic flux conditions, as the angle (\(\theta\)) affects the flux.As the door rotates, the angle changes between the magnetic field lines and the normal to the door鈥檚 surface. This angle affects the cosine component in the magnetic flux equation. For maximum flux, the angle should be zero, with the normal to the door perfectly aligned with the magnetic field lines.To reduce the magnetic flux to one-third of its maximum value, the door must rotate to a specific angle where (\(\cos(\theta) = \frac{1}{3}\)). Through trigonometric calculations, this angle is approximately (\(70.5^\circ\)). Thus, the rotation of the door changes how the magnetic field interacts with its surface, directly influencing the magnetic flux.
Trigonometric Functions
Trigonometric functions like cosine, sine, and tangent are essential tools in many physics problems, especially when dealing with angles. In this exercise, understanding the cosine function is critical.The magnetic flux through a surface is calculated using the cosine of the angle ((\(\theta\)))) between the magnetic field and the normal to the surface. This aspect of the cosine function allows for the prediction of how much magnetic flux will pass through a rotating door.Cosine values range from -1 to 1, and they measure how close the angle (\(\theta\)) moves towards maximum alignment ((\(0^\circ\)))) or complete misalignment ((\(90^\circ\)))) with the magnetic field. Here, the inverse cosine function ((\(\cos^{-1}\)))) is used to find (\(\theta\)) when you know the desired flux level relative to the maximum. It鈥檚 how we determine the specific angle the door must achieve to change the flux to precisely one-third of its maximum.

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Most popular questions from this chapter

A magnetic field is passing through a loop of wire whose area is \(0.018 \mathrm{m}^{2} .\) The direction of the magnetic field is parallel to the normal to the loop, and the magnitude of the field is increasing at the rate of \(0.20 \mathrm{T} / \mathrm{s}\) (a) Determine the magnitude of the emf induced in the loop. (b) Suppose that the area of the loop can be enlarged or shrunk. If the magnetic field is increasing as in part (a), at what rate (in \(\mathrm{m}^{2} / \mathrm{s}\) ) should the area be changed at the instant when \(B=1.8 \mathrm{T}\) if the induced emf is to be zero? Explain whether the area is to be enlarged or shrunk.

You need to design a \(60.0-\mathrm{Hz}\) ac generator that has a maximum emf of \(5500 \mathrm{V}\). The generator is to contain a 150 -turn coil that has an area per turn of \(0.85 \mathrm{m}^{2} .\) What should be the magnitude of the magnetic field in which the coil rotates?

A \(120.0-\mathrm{V}\) motor draws a current of \(7.00 \mathrm{A}\) when running at normal speed. The resistance of the armature wire is \(0.720 \Omega .\) (a) Determine the back emf generated by the motor. (b) What is the current at the instant when the motor is just turned on and has not begun to rotate? (c) What series resistance must be added to limit the starting current to \(15.0 \mathrm{A} ?\)

A \(5.40 \times 10^{-5} \mathrm{H}\) solenoid is constructed by wrapping 65 turns of wire around a cylinder with a cross-sectional area of \(9.0 \times 10^{-4} \mathrm{m}^{2} .\) When the solenoid is shortened by squeezing the turns closer together, the inductance increases to \(8.60 \times 10^{-5}\) H. Determine the change in the length of the solenoid.

The drawing shows a type of flow meter that can be used to measure the speed of blood in situations when a blood vessel is sufficiently exposed (e.g., during surgery). Blood is conductive enough that it can be treated as a moving conductor. When it flows perpendicularly with respect to a magnetic field, as in the drawing, electrodes can be used to measure the small voltage that develops across the vessel. Suppose that the speed of the blood is \(0.30 \mathrm{m} / \mathrm{s}\) and the diameter of the vessel is \(5.6 \mathrm{mm} .\) In a \(0.60-\mathrm{T}\) magnetic field what is the magnitude of the voltage that is measured with the electrodes in the drawing?
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