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Ch 27: Magnetic Field and Magnetic Forces
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 27: Magnetic Field and Magnetic ForcesProblem 13
Chapter 27, Problem 13

An open plastic soda bottle with an opening diameter of 2.5 cm is placed on a table. A uniform 1.75 T magnetic field directed upward and oriented 25° from vertical encompasses the bottle. What is the total magnetic flux through the plastic of the soda bottle?

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1
Understand the concept of magnetic flux, which is the measure of the magnetic field passing through a given area. It is calculated using the formula: \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( \Phi \) is the magnetic flux, \( B \) is the magnetic field strength, \( A \) is the area through which the field lines pass, and \( \theta \) is the angle between the magnetic field and the normal to the surface.
Calculate the area \( A \) of the opening of the soda bottle. Since the opening is circular, use the formula for the area of a circle: \( A = \pi r^2 \), where \( r \) is the radius of the circle. Given the diameter is 2.5 cm, convert it to meters and find the radius by dividing the diameter by 2.
Substitute the given values into the magnetic flux formula. Use \( B = 1.75 \text{ T} \) for the magnetic field strength, the calculated area \( A \), and \( \theta = 25^\circ \) for the angle between the magnetic field and the normal to the surface.
Convert the angle \( \theta \) from degrees to radians if necessary, as trigonometric functions in physics often use radians. Use the conversion \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \).
Calculate the magnetic flux \( \Phi \) by substituting all the known values into the formula \( \Phi = B \cdot A \cdot \cos(\theta) \) and solving for \( \Phi \). This will give you the total magnetic flux through the plastic of the soda bottle.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnetic Flux

Magnetic flux quantifies the amount of magnetic field passing through a given area. It is calculated as the product of the magnetic field strength, the area it penetrates, and the cosine of the angle between the field direction and the normal to the surface. Understanding magnetic flux is crucial for determining how magnetic fields interact with surfaces.
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Magnetic Flux

Angle of Incidence

The angle of incidence in this context refers to the angle between the magnetic field lines and the normal (perpendicular) to the surface of the bottle's opening. This angle affects the calculation of magnetic flux, as it determines how much of the field actually passes through the surface, using the cosine of the angle in the flux formula.
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Surface Area Calculation

Calculating the surface area of the bottle's opening is essential for determining the magnetic flux. For a circular opening, the area is found using the formula A = πr², where r is the radius. This area, combined with the magnetic field and angle, allows for the computation of the total magnetic flux through the bottle.
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Related Practice
Textbook Question

A flat, square surface with side length 3.40cm3.40\(\operatorname{cm}\) is in the xy-plane at z=0z = 0. Calculate the magnitude of the flux through this surface produced by a magnetic field B=(0.200T)i+(0.300T)j−(0.500T)kB=(0.200T)\(\mathbf{i}\)+(0.300T)\(\mathbf{j}\)-(0.500T)\(\mathbf{k}\).

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Textbook Question

Cyclotrons are widely used in nuclear medicine for producing short-lived radioactive isotopes. These cyclotrons typically accelerate H- (the hydride ion, which has one proton and two electrons) to an energy of 5 MeV to 20 MeV. This ion has a mass very close to that of a proton because the electron mass is negligible — about 1/2000 of the proton's mass. A typical magnetic field in such cyclotrons is 1.9 T. (a) What is the speed of a 5.0 MeV H-? (b) If the H- has energy 5.0 MeV and B = 1.9 T, what is the radius of this ion's circular orbit?

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Textbook Question

A 150 g ball containing 4.00 x 108 excess electrons is dropped into a 125 m vertical shaft. At the bottom of the shaft, the ball suddenly enters a uniform horizontal magnetic field that has magnitude 0.250 T and direction from east to west. If air resistance is negligibly small, find the magnitude and direction of the force that this magnetic field exerts on the ball just as it enters the field.

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Textbook Question

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° above the horizontal. What must the magnitude of the magnetic field be to produce a flux of 3.10 x 10-4 Wb through the surface?

Textbook Question

A deuteron (the nucleus of an isotope of hydrogen) has a mass of 3.34 x 10-27 kg and a charge of +e. The deuteron travels in a circular path with a radius of 6.96 mm in a magnetic field with magnitude 2.50 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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Textbook Question

A circular area with a radius of 6.50 cm lies in the xy-plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field B = 0.230 T at an angle of 53.1° from the +z-direction?