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28. Magnetic Fields and Forces

Magnets and Magnetic Fields

Physics

28. Magnetic Fields and Forces

Magnets and Magnetic Fields

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Problem 52b

Textbook Question

The heart produces a weak magnetic field that can be used to diagnose certain heart problems. It is a dipole field produced by a current loop in the outer layers of the heart. What is the magnitude of the heart's magnetic dipole moment?

Verified step by step guidance

Understand the concept of a magnetic dipole moment: The magnetic dipole moment ($ \mu $) is a measure of the strength of a magnetic source and is given by the formula $ \mu = I \cdot A $, where $ I $ is the current and $ A $ is the area of the current loop.

Identify the parameters: Determine the current $ I $ flowing through the loop and the area $ A $ of the loop. The area $ A $ can be calculated using $ A = \pi r^2 $, where $ r $ is the radius of the loop.

Substitute the values into the area formula: Use the given or estimated radius of the current loop to calculate $ A $ using $ A = \pi r^2 $. Ensure the radius is in meters for SI unit consistency.

Calculate the magnetic dipole moment: Substitute the values of $ I $ (current) and $ A $ (area) into the formula $ \mu = I \cdot A $. This will give the magnitude of the magnetic dipole moment in units of $ \text{A} \cdot \text{m}^2 $ (ampere-square meters).

Verify the units and interpret the result: Ensure that the units of the calculated magnetic dipole moment are consistent with $ \text{A} \cdot \text{m}^2 $. This value represents the strength of the heart's magnetic field and can be used for diagnostic purposes.

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Key Concepts

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

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that represents the strength and orientation of a magnetic source. In the context of the heart, it arises from the electrical currents generated by the heart's muscle contractions. The dipole moment is crucial for understanding how the heart's magnetic field interacts with external magnetic fields and can be measured to assess heart health.

Current Loop

A current loop is a closed loop through which electric current flows, generating a magnetic field. In the heart, the arrangement of muscle fibers and the flow of ionic currents during heartbeats create such loops. The characteristics of these loops, including their size and current strength, directly influence the resulting magnetic dipole moment of the heart.

Magnetic Field Measurement

Magnetic field measurement involves quantifying the strength and direction of magnetic fields, often using instruments like magnetometers. For the heart, techniques such as magnetocardiography can detect the weak magnetic fields produced by the heart's electrical activity. Understanding how to measure these fields is essential for diagnosing heart conditions and evaluating the heart's magnetic dipole moment.

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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?

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Textbook Question

A flat, square surface with side length $3.40 cm$ is in the xy-plane at $z equals 0$ . Calculate the magnitude of the flux through this surface produced by a magnetic field $B = (0.200 T) i + (0.300 T) j - (0.500 T) k$ .

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Textbook Question

An electron experiences a magnetic force of magnitude 4.60 x10^-15 N when moving at an angle of 60.0° with respect to a magnetic field of magnitude 3.50 x 10^-3 T. Find the speed of the electron.

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A particle with mass $1.81 \times 10^{- 3}$ $kg$ and a charge of $1.22 \times 10^{- 8} C$ has, at a given instant, a velocity $v = (3.00 \times 10^{4} m/s) j$ . What are the magnitude and direction of the particle's acceleration produced by a uniform magnetic field $B = (1.63 T) i + (0.980 T) j$ ?

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Textbook Question

A 150 V battery is connected across two parallel metal plates of area 28.5 cm² and separation 8.20 mm. A beam of alpha particles (charge +2e, mass 6.64 x 10^-27 kg) is accelerated from rest through a potential difference of 1.75 kV and enters the region between the plates perpendicular to the electric field, as shown in Fig. E27.29. What magnitude and direction of magnetic field are needed so that the alpha particles emerge undeflected from between the plates?

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Textbook Question

The Hall effect can be used to measure blood flow speed because the blood contains ions that constitute an electric current. (a) Does the sign of the ions influence the emf? Explain. (b) Determine the flow speed in an artery 3.3 mm in diameter if the measured emf across the width of the artery is 0.13 mV and B is 0.070 T. (In actual practice, an alternating magnetic field is used.)

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Textbook Question

A long copper strip is 3.0 cm wide and thick. When it carries a steady 42-A current in a 0.80-T magnetic field it produces a 6.5-μV Hall emf. Determine:

(a) the Hall field in the conductor;

(b) the drift speed of the conduction electrons;

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Textbook Question

A Hall probe, consisting of a rectangular slab of current-carrying material, is calibrated by placing it in a known magnetic field of magnitude 0.10 T. When the field is oriented normal to the slab’s rectangular face, a Hall emf of 12 mV is measured across the slab’s width. The probe is then placed in a magnetic field of unknown magnitude B, and a Hall emf of 63 mV is measured. Determine B assuming that the angle θ between the unknown field and the plane of the slab’s rectangular face is (a) θ = 90°, and (b) θ = 60°.

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