Tru Physics

Chapter 15: Magnetism

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15.1 Introduction to Magnetism

In this chapter, we will explore the fundamental principles of magnetism, the forces experienced by charged particles in a magnetic field, and the origin of magnetic fields. Understanding magnetism is essential for various applications in physics, such as motors, generators, and transformers.

Magnetism. Two bar magnets.
Bar magnets, like those pictured above, are common tools for studying magnetism. Both magnets consist of a north and south pole. If one magnet were split in half, each new piece would then still consist of a north and south pole because there are no magnetic monopoles.

15.2 Magnetic Fields

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. Magnetic fields are produced by electric currents and can exert forces on charged particles. The magnetic field is denoted by the symbol B and is measured in teslas (T). The direction of the magnetic field is always from North to South.

15.3 Magnetic Force on a Moving Charge

A charged particle moving in a magnetic field experiences a magnetic force given by:

\vec{F} = q (\vec{v} \times \vec{B})

where \vec{F} is the magnetic force, q is the charge of the particle, \vec{v} is its velocity vector, and \vec{B} is the magnetic field vector. The cross product (\times ) indicates that the direction of the magnetic force is perpendicular to both the velocity and the magnetic field. The magnitude of the magnetic force is:

F = q v B \sin(\theta)

where \theta is the angle between the velocity vector and the magnetic field vector.

15.4 Magnetic Force on a Current-Carrying Wire

A current-carrying wire in a magnetic field experiences a magnetic force given by:

\vec{F} = I\vec{L} \times \vec{B}

where \vec{F} is the magnetic force, I is the current, \vec{L} is the length vector of the wire segment, and \vec{B} is the magnetic field vector. The magnitude of the magnetic force is:

F = I L B \sin(\theta)

where \theta is the angle between the length vector of the wire segment and the magnetic field vector.

Magnetism and magnetic force due to current carrying wires.
Each of these current-carrying wires contributes to a magnetic force. Airplane designers must account for the magnitude of this force when running large bundle of wires throughout the plane.

15.5 Biot-Savart Law

The Biot-Savart law describes the magnetic field generated by a steady electric current. It states that the infinitesimal magnetic field d\vec{B} at a point P due to an infinitesimal current element d\vec{L} is given by:

d\vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{I d\vec{L} \times \hat{r}}{r^2}

where \mu_0 is the permeability of free space, I is the current, \hat{r} is the unit vector pointing from the current element to the point P, and r is the distance between the current element and point P.

15.6 Ampere’s Law

Ampere’s law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. The law is given by:

\oint \vec{B} \cdot d\vec{L} = \mu_0 I_{\text{enclosed}}

where \oint \vec{B} \cdot d\vec{L} is the line integral of the magnetic field around a closed loop, \mu_0 is the permeability of free space, and I_{\text{enclosed}} is the total current passing through the loop.

15.7 Earth’s Magnetic Field

The Earth generates a magnetic field due to the motion of electrically conducting molten iron in its outer core. The Earth’s magnetic field is roughly dipolar, with its magnetic north and south poles near the Earth’s geographic poles.

Chapter Summary

In this chapter, we covered the basic principles of magnetism, the magnetic force experienced by charged particles and current-carrying wires, the Biot-Savart law, Ampere’s law, and Earth’s magnetic field. Magnetism plays a crucial role in various applications in physics and technology, from electric motors to MRI machines.

Continue to Chapter 16: Motion in Magnetic Fields

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