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?

Charged current is$\mathrm{i}=2.50\mathrm{A}$

Wire radius is$\mathrm{R}=1.50\mathrm{mm}$

Plate radius is $\mathrm{r}=2.00\mathrm{cm}$

Here we need to use the equation for magnetic field due to charged wire and magnetic field due to charge between parallel plates.

Formula:

$\mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{ir}}{2{\mathrm{\pi R}}^2}$

$\mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{i}}{2\mathrm{\pi r}}$

Using the formula given below,

$\begin{array}{c}\mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{ir}}{2{\mathrm{\pi R}}^2}\\ =\frac{4\mathrm{\pi }\times {10}^{-7}\times 2.50\times 2.50\times {10}^{-2}}{2\mathrm{\pi }\times {(1.50\times {10}^{-3})}^2}\\ =2.22\times {10}^{-4}\mathrm{T}\\ =222\mathrm{\mu T}\end{array}$

Magnetic field at $1.00\mathrm{mm}$ inside the wire is$222\mathrm{\mu T}$

Usethefollowing formula:

$\begin{array}{c}\mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{i}}{2\mathrm{\pi r}}\\ =\frac{4\mathrm{\pi }\times {10}^{-7}\times 2.50}{2\mathrm{\pi }\times 3\times {10}^{-3}}\\ =1.667\times {10}^{-4}\mathrm{T}\\ =167\mathrm{\mu T}\end{array}$

Magnetic field at $3.00\mathrm{mm}$ outside the wire is $167\mathrm{\mu T}$

$\begin{array}{c}\mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{i}}{2\mathrm{\pi r}}\\ =\frac{4\mathrm{\pi }\times {10}^{-7}\times 2.50}{2\mathrm{\pi }\times 2.20\times {10}^{-2}}\\ =2.27\times {10}^{-5}\mathrm{T}\\ =22.7\times {10}^{-6}\mathrm{T}\\ =22.7\mathrm{\mu T}\end{array}$

Magnetic field at $2.20\mathrm{cm}$ outside the wire is $22.7\mathrm{\mu T}$

Use the following formula:

$\begin{array}{c}\mathrm{B}=\frac{{\mathrm{\mu }}_0{\mathrm{i}}_{\mathrm{d}}\mathrm{r}}{2{\mathrm{\pi R}}^2}\\ =\frac{4\mathrm{\pi }\times {10}^{-7}\times 2.50\times 1\times {10}^{-3}}{2\mathrm{\pi }\times {(2\times {10}^{-2})}^2}\\ =1.25\times {10}^{-6}\mathrm{T}\\ =1.25\mathrm{\mu T}\end{array}$

Magnetic field due to $i_d$ at $1.00\mathrm{mm}$ inside the gap is $1.25\mathrm{\mu T}$

Use the following formula:

$\begin{array}{c}\mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{ir}}{2{\mathrm{\pi R}}^2}\\ =\frac{4\mathrm{\pi }\times {10}^{-7}\times 2.50\times 3\times {10}^{-3}}{2\mathrm{\pi }\times {(2\times {10}^{-2})}^2}\\ =3.75\times {10}^{-6}\mathrm{T}\\ =3.75\mathrm{\mu T}\end{array}$

Magnetic field due to $i_d$ at $3.00$ mm inside the gap is $3.75\mathrm{\mu T}$

Use the following formula:

$\begin{array}{c}\mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{ir}}{2{\mathrm{\pi R}}^2}\\ =\frac{4\mathrm{\pi }\times {10}^{-7}\times 2.50}{2\mathrm{\pi }\times (2.2\times {10}^{-2})}\\ =2.27\times {10}^{-5}\mathrm{T}\\ =22.7\mathrm{\mu T}\end{array}$

Magnetic field due to role="math" localid="1662983978538" ${\mathrm{i}}_{\mathrm{d}}$ at$2.20cm$ outside the gap is $22.7\mathrm{\mu T}$

Why the field at two smaller radii is so different?

Because displacement current in the gap is spread over a larger cross-sectional area, values of B within that area are relatively small. Outside that cross section, the two values of B are identical.