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The capacitor in Fig. 32-7 is being charged with a $\mathbf{2}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{A}$ current. The wire radius is $\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{mm}$, and the plate radius is$\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{cm}$ . Assume that the current i in the wire and the displacement current id in the capacitor gap are both uniformly distributed. What is the magnitude of the magnetic field due to i at the following radial distances from the wire’s center: (a)$\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mm}$ (inside the wire), (b) $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mm}$(outside the wire), and (c) role="math" localid="1662982620568" $\mathbf{2}\mathbf{.}\mathbf{20}\mathbf{}\mathbf{cm}$(outside the wire)? What is the magnitude of the magnetic field due to id at the following radial distances from the central axis between the plates: (d) $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mm}$(inside the gap), (e)$\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{mm}$ (inside the gap), and (f) (outside the gap)? (g)$\mathbf{2}\mathbf{.}\mathbf{20}\mathbf{}\mathbf{cm}$ Explain why the fields at the two smaller radii are so different for the wire and the gap but the fields at the largest radius are not?

A parallel-plate capacitor with circular plates of radius $\mathbf{R}\mathbf{=}\mathbf{16}\mathbf{}\mathbf{mm}$and gap width$\mathbf{d}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mm}$has a uniform electric field between the plates. Starting at time$\mathbf{t}\mathbf{=}\mathbf{0}$, the potential difference between the two plates is$\mathbf{V}\mathbf{=}\mathbf{\left(}\mathbf{100}\mathbf{V}\mathbf{\right)}{\mathbf{e}}^{\mathbf{−}\frac{\mathbf{t}}{\mathbf{Ï„}}}$, where the time constant$\mathbf{Ï„}\mathbf{=}\mathbf{12}\mathbf{}\mathbf{ms}$. At radial distance$\mathbf{r}\mathbf{=}\mathbf{0}\mathbf{.8}\mathbf{}\mathbf{R}$from the central axis, what is the magnetic field magnitude (a) as a function of time for$\mathbf{t}\mathbf{â‰\yen }\mathbf{0}$and (b) at time$\mathbf{t}\mathbf{=}\mathbf{3}\mathbf{Ï„}$?

The induced magnetic field at a radial distance 6.0 mm from the central axis of a circular parallel-plate capacitor is $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{×}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{}\mathit{T}$. The plates have a radius 3.0 mm. At what rate$\mathit{d}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{E}}\mathbf{/}\mathit{d}\mathit{t}$is the electric field between the plates changing?

An electron in an external magnetic field ${\stackrel{\mathbf{â‡\P Ä}}{\mathbf{B}}}_{\mathbf{e}\mathbf{x}\mathbf{t}}$. has its spin angular momentum S_zantiparallel to ${\stackrel{\mathbf{â‡\P Ä}}{\mathbf{B}}}_{\mathbf{ext}}$. If the electron undergoes a spin-flip so thatS_z is then parallel th ${\stackrel{\mathbf{â‡\P Ä}}{\mathbf{B}}}_{\mathbf{ext}}$, must energy be supplied to or lost by the electron?

A magnetic flux of$\mathbf{7}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mWb}$ is directed outward through the flat bottom face of the closed surface shown in Fig. 32-40. Along the flat top face (which has a radius of$\mathbf{4}\mathbf{.}\mathbf{2}\mathbf{}\mathbf{cm}$ ) there is a $\mathbf{0}\mathbf{.}\mathbf{40}\mathbf{}\mathbf{T}$magnetic field directed perpendicular to the face. What are the (a) magnitude and (b) direction (inward or outward) of the magnetic flux through the curved part of the surface?

The magnetic field of Earth can be approximated as the magnetic field of a dipole. The horizontal and vertical components of this field at any distance r from Earth’s center are given by ${\mathbf{B}}_{\mathbf{H}}\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}\mathbf{μ}}{\mathbf{4}{\mathbf{Ï€°ù}}^{\mathbf{3}}}\mathbf{×}{\mathbf{³¦´Ç²õλ}}_{\mathbf{m}}\mathbf{}$,${\mathbf{B}}_{\mathbf{v}}\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}\mathbf{μ}}{\mathbf{2}{\mathbf{Ï€°ù}}^{\mathbf{3}}}\mathbf{×}{\mathbf{²õ¾\pm ²Ôλ}}_{\mathbf{m}}$where l_m is the magnetic latitude (this type of latitude is measured from the geomagnetic equator toward the north or south geomagnetic pole). Assume that Earth’s magnetic dipole moment has magnitude$\mathbf{μ}\mathbf{=}\mathbf{}\mathbf{}\mathbf{8.00}\mathbf{}\mathbf{×}\mathbf{}\mathbf{1022}\mathbf{}\mathbf{A}\mathbf{}{\mathbf{m}}^{\mathbf{2}}$ . (a) Show that the magnitude of Earth’s field at latitude lm is given by$\mathbf{B}\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}\mathbf{μ}}{\mathbf{4}{\mathbf{Ï€°ù}}^{\mathbf{3}}}\mathbf{×}\sqrt{\mathbf{1}\mathbf{+}\mathbf{3}{\mathbf{sin}}^{\mathbf{2}}{\mathbf{λ}}_{\mathbf{m}}\mathbf{}\mathbf{}}$

A parallel-plate capacitor with circular plates of radius is being charged. At what radius (a) inside and (b) outside the capacitor gap is the magnitude of the induced magnetic field equal to$\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{\%}$of its maximum value?

A sample of the paramagnetic salt to which the magnetization curve of Fig. 32-14 applies is immersed in a uniform magnetic field of $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{\text{T}}$. At what temperature will the degree of magnetic saturation of the sample be (a)$\mathbf{50}\mathbf{\%}$and (b)$\mathbf{90}\mathbf{\%}$
?

A parallel-plate capacitor with circular plates of radius $\mathit{R}$is being discharged. The displacement current through a central circular area, parallel to the plates and with radius $\raisebox{1ex}{$\mathbf{R}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.$, is $\mathbf{\text{2.0A}}$. What is the discharging current?

Figure 32-41 gives the variation of an electric field that is perpendicular to a circular area of ${\mathbf{\text{2.0m}}}^{\mathbf{\text{2}}}$. During the time period shown, what is the greatest displacement current through the area?