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Chapter 22: Problem 15

A magnetic field has a magnitude of \(0.078 \mathrm{T}\) and is uniform over a circular surface whose radius is \(0.10 \mathrm{m}\). The field is oriented at an angle of \(\phi=25^{\circ}\) with respect to the normal to the surface. What is the magnetic flux through the surface?

Short Answer

Expert verified
The magnetic flux through the surface is approximately 0.00222 Wb.

Step by step solution

01

Define Magnetic Flux Formula

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

Calculate the Area of the Circular Surface

The area \( A \) of a circle is calculated using the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle. Here, \( r = 0.10 \) m. So, \( A = \pi \times (0.10)^2 \approx 0.0314 \) m虏.
03

Insert Values into the Magnetic Flux Formula

Substitute \( B = 0.078 \) T, \( A = 0.0314 \) m虏, and \( \phi = 25^{\circ} \) into the flux formula: \( \Phi = 0.078 \cdot 0.0314 \cdot \cos(25^{\circ}) \).
04

Calculate \( \cos(25^{\circ}) \)

Find the cosine of 25 degrees. \( \cos(25^{\circ}) \approx 0.9063 \).
05

Compute the Magnetic Flux

Now multiply the values: \( \Phi = 0.078 \cdot 0.0314 \cdot 0.9063 \approx 0.00222 \). Therefore, the magnetic flux through the surface is approximately 0.00222 Weber (Wb).

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Key Concepts

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

Magnetic Field
Understanding a magnetic field is crucial. A magnetic field is a region around a magnetic material or a moving electric charge where the force of magnetism acts. It is represented by the symbol \( B \) and its strength or magnitude is measured in Teslas (T). In our example, we have a uniform magnetic field of magnitude \( 0.078 \, \mathrm{T} \). This uniformity means its strength is the same throughout the surface area we're considering.
Magnetic fields affect objects with magnetic properties and are used in various technologies such as MRI machines and electric motors. In the context of this problem, the magnetic field interacts with a circular surface to determine the magnetic flux, which is a measure of the flow of the magnetic field through a given area.
Circular Surface Area
The circular surface area plays a key role in calculating magnetic flux. To find the area \( A \) of a circle, we use the mathematical formula \( A = \pi r^2 \), where \( r \) is the radius.
For our specific problem, the circular surface has a radius of \( 0.10 \, \mathrm{m} \). By substituting this value into the area formula, we compute \( A = \pi \times (0.10)^2 \approx 0.0314 \, \mathrm{m}^2 \).
Understanding how to calculate the area of a circle is essential in many areas of physics and engineering, as it allows us to determine the extent of the surface through which magnetic lines pass.
Angle of Inclination
The angle of inclination \( \phi \) is an essential factor when examining how a magnetic field interacts with a surface. It is the angle between the direction of the magnetic field and the line perpendicular (normal) to the surface in question.
In our problem, this angle is \( 25^{\circ} \). The importance of this angle cannot be overstated as it influences the effective component of the magnetic field passing through the surface.
Recognizing the significance of this angle helps in understanding how max and min magnetic influences are dictated by this angular relationship. A \( 0^{\circ} \) angle means maximum flux, while a \( 90^{\circ} \) angle means zero flux as the field lines run parallel to the surface and do not penetrate it.
Cosine Function
The cosine function is fundamental in calculating the magnetic flux through a surface when a magnetic field is applied. This trigonometric function, denoted as \( \cos(\phi) \), gives the ratio of the adjacent side to the hypotenuse in a right triangle. It helps transform the angle of inclination into a scalar that affects the magnetic field's magnitude.
For an angle \( \phi = 25^{\circ} \), \( \cos(25^{\circ}) \approx 0.9063 \). This value represents the proportion of the magnetic field that effectively contributes to the flux through the surface.
  • If \( \phi = 0^{\circ} \), \( \cos(\phi) = 1 \) and all of the magnetic field goes through.
  • If \( \phi = 90^{\circ} \), \( \cos(\phi) = 0 \) and none passes through the surface.
Understanding this function is vital as it plays a crucial role in the magnetic flux formula, directly influencing the outcome of our calculations.

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Most popular questions from this chapter

A loop of wire has the shape shown in the drawing. The top part of the wire is bent into a semicircle of radius \(r=0.20 \mathrm{m}\). The normal to the plane of the loop is parallel to a constant magnetic field \(\left(\phi=0^{\circ}\right)\) of magnitude 0.75 T. What is the change \(\Delta \Phi\) in the magnetic flux that passes through the loop when, starting with the position shown in the drawing, the semicircle is rotated through half a revolution?

A rectangular loop of wire with sides 0.20 and \(0.35 \mathrm{m}\) lies in a plane perpendicular to a constant magnetic field (see part \(a\) of the drawing). The magnetic field has a magnitude of \(0.65 \mathrm{T}\) and is directed parallel to the normal of the loop's surface. In a time of 0.18 s, one-half of the loop is then folded back onto the other half, as indicated in part \(b\) of the drawing. Determine the magnitude of the average emf induced in the loop.

Magnetic resonance imaging (MRI) is a medical technique for producing pictures of the interior of the body. The patient is placed within a strong magnetic field. One safety concern is what would happen to the positively and negatively charged particles in the body fluids if an equipment failure caused the magnetic field to be shut off suddenly. An induced emf could cause these particles to flow, producing an electric current within the body. Suppose the largest surface of the body through which flux passes has an area of \(0.032 \mathrm{m}^{2}\) and a normal that is parallel to a magnetic field of \(1.5 \mathrm{T}\). Determine the smallest time period during which the field can be allowed to vanish if the magnitude of the average induced emf is to be kept less than \(0.010 \mathrm{V}\).

A generator uses a coil that has 100 turns and a \(0.50-\mathrm{T}\) magnetic field. The frequency of this generator is \(60.0 \mathrm{Hz},\) and its emf has an rms value of \(120 \mathrm{V}\). Assuming that each turn of the coil is a square (an approximation), determine the length of the wire from which the coil is made.

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.
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