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10.md

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    Erik Strand authored
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    title: Problem Set 10

    (13.1)

    (a)

    {:.question} Estimate the diamagnetic susceptibility of a typical solid.

    Starting from equation 12.15,

    There was an error rendering this math block. KaTeX parse error: Undefined control sequence: \num at position 67: … m_e V} \\ &= -\̲n̲u̲m̲{1.26e-6} \si{N…

    (b)

    {:.question} Using this, estimate the field strength needed to levitate a frog, assuming a gradient that drops to zero across the frog. Express your answer in teslas.

    From 12.7,

    F=Vμ0χmHdHdz F = -V \mu_0 \chi_m H \frac{d H}{d z}

    I'll assume the frog is 0.1 meters tall, has a mass of 0.1 kg, and a volume of

    10^{-4} \si{m^3}
    (this is consistent with the frog being mostly water). I'll also assume the magnetic field gradient is constant, so
    dH/dz = H / 0.1
    . Solving for
    H
    ,

    \begin{align*} H &= \sqrt{-\frac{F z}{V \mu_0 \chi_m}} \\ &= \sqrt{-\frac{0.1 \si{kg} \cdot 9.8 \si{m/s^2} \cdot 0.1 \si{m}} {10^{-4} \si{m^3} \cdot \num{1.26e-6} \si{N/A^2} \cdot \num{-8.9e-5}}} \\ &= \num{3e6} \si{A/m} \\ \end{align*}

    Thus the magnetic field is

    \begin{align*} B &= \mu_0 H \\ &= \num{1.26e-6} \si{N/A^2} \cdot \num{3e6} \si{A/m} \\ &= 3.7 \si{T} \end{align*}

    (13.2)

    {:.question} Estimate the size of the direct magnetic interaction energy between two adjacent free electrons in a solid, and compare this to the size of their electrostatic interaction energy. Remember that the field of a magnetic dipole

    \vec{m}
    is

    \vec{B} = \frac{\mu_0}{4 \pi} \left[ \frac{3 \hat{x} (\hat{x} \cdot \vec{m}) - \vec{m}}{|\vec{x}|^3} \right]

    (13.3)

    {:.question} Using the equation for the energy in a magnetic field, describe why:

    (a)

    {:.question} A permanent magnet is attracted to an unmagnetized ferromagnet.

    (b)

    {:.question} The opposite poles of permanent magnets attract each other.

    (13.4)

    {:.question} Estimate the saturation magnetization for iron at 0 K.

    (13.5)

    (a)

    {:.question} Show that the area enclosed in a hysteresis loop in the

    (B,H)
    plane is equal to the energy dissipated in going around the loop.

    (b)

    {:.question} Estimate the power dissipated if 1 kg of iron is cycled through a hysteresis loop at 60 Hz; the coercivity of iron is

    \num{4e3} \si{A/m}
    .

    (13.6)

    {:.question} Approximately what current would be required in a straight wire to be able to erase a

    \gamma \text{-} Fe_2 O_3
    recording at a distance of 1 cm?