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Chapter 27: Problem 12

A horizontal rectangular surface has dimensions 2.80 cm by 3.20 cm and is in a uniform magnetic field that is directed at an angle of 30.0\(^\circ\) above the horizontal. What must the magnitude of the magnetic field be to produce a flux of 3.10 \(\times\) 10\(^{-4}\) Wb through the surface?

Short Answer

Expert verified
The magnetic field magnitude must be approximately 0.400 T.

Step by step solution

01

Understand the Magnetic Flux Formula

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

Calculate Area of the Surface

The area \( A \) of the rectangle can be calculated using the formula for the area of a rectangle: \( A = \text{length} \times \text{width} = 2.80 \text{ cm} \times 3.20 \text{ cm} = 8.96 \text{ cm}^2 \). Convert the area to square meters for consistency in units: \( 8.96 \text{ cm}^2 = 8.96 \times 10^{-4} \text{ m}^2 \).
03

Set Up the Equation for Magnetic Flux

Using the magnetic flux formula from Step 1, we have: \( 3.10 \times 10^{-4} \text{ Wb} = B \cdot 8.96 \times 10^{-4} \text{ m}^2 \cdot \cos(30.0^\circ) \).
04

Solve for Magnetic Field Magnitude \( B \)

Rearrange the magnetic flux equation to solve for \( B \): \( B = \frac{3.10 \times 10^{-4} \text{ Wb}}{8.96 \times 10^{-4} \text{ m}^2 \times \cos(30.0^\circ)} \).
05

Calculate \( \cos(30.0^\circ) \)

The value of \( \cos(30.0^\circ) \) is \( \sqrt{3}/2 \approx 0.866 \).
06

Substitute and Compute \( B \)

Substitute the values into the rearranged equation: \( B = \frac{3.10 \times 10^{-4}}{8.96 \times 10^{-4} \times 0.866} \approx 0.400 \text{ T} \).

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Key Concepts

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

Magnetic Field
A magnetic field refers to a field surrounding a magnetic material or a moving electric charge within which the force of magnetism acts. It is denoted by the symbol \( B \) and is measured in Tesla (T). Magnetic fields are fundamental to understanding the behavior of magnetic flux, which pertains to the quantity of magnetism, represented as magnetic field lines, passing through a given surface. This field can be uniform, meaning it has the same strength and direction at all points in the area it covers.

In the context of this exercise, we are dealing with a uniform magnetic field, which simplifies the calculations significantly. The strength of the magnetic field is determined, so we measure how much magnetic flux passes through the rectangular surface when the field is not perpendicular to the surface. The calculation of the magnetic flux involves the angle between the magnetic field direction and the normal (perpendicular) vector to the surface, which, in this case, is what causes the need to use the cosine of the angle. Understanding this angle's effect is crucial for effectively using trigonometric functions in the calculations.
Surface Area Calculation
Calculating surface area is essential in determining how much magnetic flux passes through a given surface. For a rectangle, such as the one in this exercise, the area \( A \) can be calculated by multiplying the length by the width of the rectangle.

Here, the rectangle measures 2.80 cm by 3.20 cm, leading to an area of \( 8.96 \text{ cm}^2 \). Since scientific calculations often require uniform units, we convert this area into square meters: \( 8.96 \times 10^{-4} \text{ m}^2 \). This standard unit ensures consistent and accurate calculations within the framework of the magnetic flux formula. Recognizing how these calculations are interconnected is key for solving problems involving magnetic fields.
Trigonometric Functions
Trigonometric functions play a vital role in calculating magnetic flux because they help determine the component of the magnetic field that is perpendicular to the surface. In this instance, the angle between the magnetic field and the surface’s normal is 30 degrees. We need to use the cosine of this angle, as the magnetic flux formula includes the term \( \cos(\theta) \), representing this perpendicular component.

The cosine of a 30-degree angle is \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 \). This value is essential when plugged into the flux formula, showing how much of the magnetic field effectively penetrates the surface area.

Understanding and applying trigonometric functions accurately links directly to calculating the results in magnetic scenarios and beyond, as angles often adjust how fields impact surfaces.

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Most popular questions from this chapter

An electron moves at 1.40 \(\times\) 10\(^6\) m/s through a region in which there is a magnetic field of unspecified direction and magnitude 7.40 \(\times\) 10\(^{-2}\) T. (a) What are the largest and smallest possible magnitudes of the acceleration of the electron due to the magnetic field? (b) If the actual acceleration of the electron is one-fourth of the largest magnitude in part (a), what is the angle between the electron velocity and the magnetic field?

A mass spectrograph is used to measure the masses of ions, or to separate ions of different masses (see Section 27.5). In one design for such an instrument, ions with mass \(m\) and charge \(q\) are accelerated through a potential difference \(V\). They then enter a uniform magnetic field that is perpendicular to their velocity, and they are deflected in a semicircular path of radius \(R\). A detector measures where the ions complete the semicircle and from this it is easy to calculate \(R\). (a) Derive the equation for calculating the mass of the ion from measurements of \(B\), \(V\), \(R\), and \(q\). (b) What potential difference \(V\) is needed so that singly ionized \(^{12}\)C atoms will have \(R =\) 50.0 cm in a 0.150-T magnetic field? (c) Suppose the beam consists of a mixture of \(^{12}\)C and \(^{14}\)C ions. If \(v\) and \(B\) have the same values as in part (b), calculate the separation of these two isotopes at the detector. Do you think that this beam separation is sufficient for the two ions to be distinguished? (Make the assumption described in Problem 27.59 for the masses of the ions.)

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\(^6\) m/s in a circular orbit of radius 5.3 \(\times\) 10\(^{-11}\) 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?

The amount of meat in prehistoric diets can be determined by measuring the ratio of the isotopes \(^{15}\)N to \(^{14}\)N in bone from human remains. Carnivores concentrate \(^{15}\)N, so this ratio tells archaeologists how much meat was consumed. For a mass spectrometer that has a path radius of 12.5 cm for \(^{12}\)C ions (mass 1.99 \(\times\) 10\(^{-26}\) kg), find the separation of the \(^{14}\)N 1mass 2.32 \(\times\) 10\(^{-26}\) kg2 and 15N (mass 2.49 \(\times\) 10\(^{-26}\) kg) isotopes at the detector.

A particle with negative charge q and mass \(m =\) 2.58 \(\times\) 10\(^{-15}\) kg is traveling through a region containing a uniform magnetic field \(\overrightarrow{B} =\) -(0.120 T)\(\hat{k}\). At a particular instant of time the velocity of the particle is \(\vec{v}\) (1.05 \(\times\) 10\(^6\) m/s (-3\(\hat{\imath}\)+4\(\hat{\jmath}\)+12\(\hat{k}\)) and the force \(\overrightarrow{F}\) on the particle has a magnitude of 2.45 N. (a) Determine the charge \(q\). (b) Determine the acceleration \(\overrightarrow{a}\) of the particle. (c) Explain why the path of the particle is a helix, and determine the radius of curvature \(R\) of the circular component of the helical path. (d) Determine the cyclotron frequency of the particle. (e) Although helical motion is not periodic in the full sense of the word, the \(x\)- and \(y\)-coordinates do vary in a periodic way. If the coordinates of the particle at \(t =\) 0 are (\(x, y, z\)) = (\(R\), 0, 0), determine its coordinates at a time \(t =\) 2\(T\), where \(T\) is the period of the motion in the \(xy\)-plane.
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