Problems

The figure gives the magnetization curve for a paramagnetic material. The vertical axis scale is set by $a=0.15,$ and the horizontal axis scale is set by $b=0.2\mathrm{T} {\mathrm{K}}^{-1}$. Let ${\mu }_{\text{sam }}$ be the measured net magnetic moment of a sample of the material and ${\mu }_{\max }$ be the maximum possible net magnetic moment of that sample. According to Curie's law, what would be the ratio ${\mu }_{sam}/{\mu }_{\max }$ were the sample placed in a uniform magnetic field of magnitude $0.800 \mathrm{T}$, at a temperature of $3.80 \mathrm{K}?$

In Fig., a uniform electric field $\overrightarrow{E}$ collapses. The vertical axis scale is set by $E_s=6.0\times {10}^5 \mathrm{N} {\mathrm{C}}^{-1}$ and the horizontal axis scale is set by $t_s=24.0 \mathrm{\mu s}$. Calculate the magnitude of the displacement current through a $1.6 {\mathrm{m}}^2$ area perpendicular to the field during each of the time intervals $a, b,$ and $c$ shown on the graph. (Ignore the behavior at the ends of the intervals.)

Two wires, parallel to a z-axis and a distance $4r$ apart, carry equal currents $i$ in opposite directions, as shown in the figure. A circular cylinder of radius $r$ and length $L$ has its axis on the $z$-axis, midway between the wires. (a) Use Gauss' law for magnetism to derive an expression for the net outward magnetic flux through the half of the cylindrical surface above the $x$-axis. (Hint: Find the flux through the portion of the $x$-z plane that lies within the cylinder). (b) Evaluate the flux for current $i=2.40 \mathrm{A}$ and length $L=3.40 \mathrm{cm}$.

What is the measured component of the orbital magnetic dipole moment of an electron with (a) $m_l=1$ and $\left(b\right)m_l=-3?$

A sample of the paramagnetic salt to which the magnetization curve of the figure applies is held at room temperature $(300 \mathrm{K})$. At what applied magnetic field will the degree of magnetic saturation of the sample be $\left(a\right)$ $65\%$ and
$\left(b\right)$ $90\%?$ $\left(c\right)$ Are these fields attainable in the laboratory? 

The magnetic flux through each of five faces of a die (singular of dice) is given by ${\Phi }_{\mathrm{B}}=\pm 2N \mathrm{Wb}$, where $N(=1$ to $5)$ is the number of spots on the face. The flux is positive (outwards) for $N$ even and negative (inward) for $N$ odd. What is the flux through the sixth face of the die?

A parallel-plate capacitor with circular plates of radius $40 \mathrm{mm}$ is being discharged by a current of $6.0 \mathrm{A}$. At what radius
$\left(a\right)$ inside and

$\left(b\right)$ outside the capacitor gap is the magnitude of the induced magnetic field equal to $60\%$ of its maximum value?

$\left(c\right)$ What is that maximum value?

In the figure, a parallel-plate capacitor has square plates of edge length $L=1.0 \mathrm{m}$. A current of $3.0 \mathrm{A}$ charges the capacitor, producing a uniform electric field $\overrightarrow{E}$ between the plates, with $\overrightarrow{E}$ perpendicular to the plates. $\left(a\right)$ What is the displacement current $i_{\mathrm{d}}$ through the region between the plates? $\left(b\right)$ What is $\frac{dE}{dt}$ in this region? $\left(c\right)$ What is the displacement current encircled by the square dashed path of edge length $d=0.50 \mathrm{m}$? $\left(d\right)$ What is the value of $\oint \overrightarrow{B}·\mathrm{d}\overrightarrow{s}$ around this square dashed path?