Chapter 9 Review

Key Terms

Ampère’s law
physical law that states that the line integral of the magnetic field around an electric current is proportional to the current

Biot-Savart law
an equation giving the magnetic field at a point produced by a current-carrying wire

diamagnetic materials
their magnetic dipoles align oppositely to an applied magnetic field; when the field is removed, the material is unmagnetized

ferromagnetic materials
contain groups of dipoles, called domains, that align with the applied magnetic field; when this field is removed, the material is still magnetized

hysteresis
property of ferromagnets that is seen when a material’s magnetic field is examined versus the applied magnetic field; a loop is created resulting from sweeping the applied field forward and reverse

magnetic domains
groups of magnetic dipoles that are all aligned in the same direction and are coupled together quantum mechanically

magnetic susceptibility
ratio of the magnetic field in the material over the applied field at that time; positive susceptibilities are either paramagnetic or ferromagnetic (aligned with the field) and negative susceptibilities are diamagnetic (aligned oppositely with the field)

paramagnetic materials
their magnetic dipoles align partially in the same direction as the applied magnetic field; when this field is removed, the material is unmagnetized

permeability of free space

$\mu_0,$

measure of the ability of a material, in this case free space, to support a magnetic field

solenoid
thin wire wound into a coil that produces a magnetic field when an electric current is passed through it

toroid
donut-shaped coil closely wound around that is one continuous wire

Key Equations

Permeability of free space	$\mu_0=4\pi\times10^{-7}~\mathrm{T}\cdot\mathrm{m/A}$
Contribution to magnetic field from a current element	$dB=\frac{\mu_0}{4\pi}\frac{Idlsin\theta}{r^2}$
Biot–Savart law	$\vec{\mathbf{B}}=\frac{\mu_0}{4\pi}\int_{\mathrm{wire}}\frac{Id\vec{\mathbf{l}}\times\hat{\mathbf{r}}}{r^2}$
Magnetic field due to a long straight wire	$B=\frac{\mu_0I}{2\pi R}$
Force between two parallel currents	$\frac{F}{l}=\frac{\mu_0I_1I_2}{2\pi r}$
Magnetic field of a current loop	$B=\frac{\mu_0I}{2R} (at Center of loop)$
Ampère’s law	$\oint\vec{\mathbf{B}}\cdot\vec{\mathbf{dl}}=\mu_0I$
Magnetic field strength inside a solenoid	$B=\mu_0nI$
Magnetic field strength inside a toroid	$B=\frac{\mu_0NI}{2\pi r}$
Magnetic permeability	$\mu=(1+\chi)\mu_0$
Magnetic field of a solenoid filled with paramagnetic material Magnetic field due to a	$B=\mu nI$

Summary

9.1 The Biot-Savart Law

The magnetic field created by a current-carrying wire is found by the Biot-Savart law.
The current element $I\vec{\mathbf{dl}}$ produces a magnetic field a distance $r$ away.

9.2 Magnetic Field Due to a Thin Straight Wire

The strength of the magnetic field created by current in a long straight wire is given by $B=\mu_0nI$ (long straight wire) where $I$ is the current, $R$ is the shortest distance to the wire, and the constant $\mu_0=4\pi\times10^{-7}~\mathrm{T}\cdot\mathrm{m/s}$ is the permeability of free space.
The direction of the magnetic field created by a long straight wire is given by right-hand rule $2$ (RHR $2$ ): Point the thumb of the right hand in the direction of current, and the fingers curl in the direction of the magnetic field loops created by it.

9.3 Magnetic Force between Two Parallel Currents

The force between two parallel currents $I_1$ and $I_2,$ separated by a distance $r,$ has a magnitude per unit length given by $\frac{F}{l}=\frac{\mu_0I_1I_2}{2\pi r}.$
The force is attractive if the currents are in the same direction, repulsive if they are in opposite directions.

9.4 Magnetic Field of a Current Loop

The magnetic field strength at the center of a circular loop is given by $B=\frac{\mu_0I}{2R}$ (at center of loop), where $R$ is the radius of the loop. RHR-2 gives the direction of the field about the loop.

9.5 Ampère’s Law

The magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field known as Ampère’s law.
Ampère’s law can be used to determine the magnetic field from a thin wire or thick wire by a geometrically convenient path of integration. The results are consistent with the Biot-Savart law.

9.6 Solenoids and Toroids

The magnetic field strength inside a solenoid is

(inside a solenoid)

where

is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.

The magnetic field strength inside a toroid is

(within the toroid)

where

is the number of windings. The field inside a toroid is not uniform and varies with the distance as

9.7 Magnetism in Matter

Materials are classified as paramagnetic, diamagnetic, or ferromagnetic, depending on how they behave in an applied magnetic field.
Paramagnetic materials have partial alignment of their magnetic dipoles with an applied magnetic field. This is a positive magnetic susceptibility. Only a surface current remains, creating a solenoid-like magnetic field.
Diamagnetic materials exhibit induced dipoles opposite to an applied magnetic field. This is a negative magnetic susceptibility.
Ferromagnetic materials have groups of dipoles, called domains, which align with the applied magnetic field. However, when the field is removed, the ferromagnetic material remains magnetized, unlike paramagnetic materials. This magnetization of the material versus the applied field effect is called hysteresis.

Answers to Check Your Understanding

9.1

9.2

9.3 4 amps flowing out of the page

9.4 Both have a force per unit length of

9.5

9.6 In these cases the integrals around the Ampèrian loop are very difficult because there is no symmetry, so this method would not be useful.

9.7 a.

9.8 a.