Question

The figure 32-20 shows a circular region of radius $\mathit{R}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{cm}$in which adisplacement currentis directedout of the page. The magnitude of the density of this displacement current is ${\mathit{J}}_{\mathbf{d}}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{A}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{r}\mathbf{/}\mathbf{R}\mathbf{)}$, where $\mathit{r}$ is the radial distance $\mathit{r}\mathbf{\le }\mathit{R}$. (a) What is the magnitude of the magnetic field due to displacement current at $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{cm}$? (b)What is the magnitude of the magnetic field due to displacement current at $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{cm}$?

Fig 32-20

Short answer

(a) The magnitude of the magnetic field due to displacement current at a radial distance at $R=2.00\mathrm{cm}$is $B=27.9\mathrm{nT}$.

(b) The magnitude of the magnetic field due to displacement current at a radial distance at $R=5.00\mathrm{cm}$is $B=15.1\mathrm{nT}$.

Step-by-step solution

The given data

a) Displacement current density, $J_d=\left(4.00\raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}^2$}\right.\right)\left(1-\raisebox{1ex}{$r$}\!\left/ \!\raisebox{-1ex}{$R$}\right.\right)$

b) The radius of the circular region, $R=3.00\text{cm}\times \frac{1\text{m}}{100\text{cm}}=3.00\times {10}^{-2}\text{m}$

c) Radial distances at which the magnetic field is induced, $r_1=2\text{cm}\times \frac{1}{100\text{m}}=0.02\text{m}$, $r_2=5\text{cm}\times \frac{1}{100\text{m}}=0.05\text{m}$

Understanding the concept of induced magnetic field

When a conductor is placed in a region of changing magnetic field, it induces a displacement current that starts flowing through it as it causes the case of an electric field produced in the conductor region. According to Lenz law, the current flows through the conductor to oppose the change in magnetic flux through the area enclosed by the loop or the conductor. The magnitude of the magnetic field is due to the displacement current using the displacement current density, which is non-uniformly distributed.

Formulae:

The magnetic field at a point inside the capacitor, $B=\left(\frac{{\mu }_0{i}_{d}r}{2\pi R^2}\right)$.....(i)

, where $B$is the magnetic field, ${\mu }_0=\left(4\pi \times {10}^{-7}\text{T.m/A}\right)$is the magnetic permittivity constant, $r$is the radial distance, $i_d$is the displacement current, $R$is the radius of the circular region.

The magnetic field at a point outside the capacitor, $B=\left(\frac{{\mu }_0{i}_d}{2\pi r}\right)$.....(ii)

Where, ${\mu }_0=\left(4\pi \times {10}^{-7}\text{T.m/A}\right)$is the magnetic permittivity constant, $r$is the radial distance, $i_d$is the displacement current.

The current flowing in a given region for a non-uniform electric field, $i_{d,enc}={\int }_0^{r}J\left(2\mathrm{\pi r}\right)dr$......(iii)

Where, $J$is the current density of the material, $r$is the radial distance of the circular region, $dr$is the differential form of the radial distance.

(a) Determining the magnitude of the magnetic field due to displacement current at a radial distance R=2.00 cm.

The displacement current density is non-uniform. Hence, the displacement current is determined by taking the integration over the closed path of radius $r$,$r_1=0.02\text{m}$, $r_1<R$and that is given using the given data in equation (i) as follows:

role="math" localid="1663225180304" $\begin{array}{rcl}{i}_{d,enc}& =& {\int }_0^r\left(4.00\raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}^2$}\right.\right)\left(1-\raisebox{1ex}{$r$}\!\left/ \!\raisebox{-1ex}{$R$}\right.\right)\left(2\mathrm{\pi r}\right)dr\\ & =& \left(8\mathrm{\pi }\raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}^2$}\right.\right){\int }_0^r\left(r-\raisebox{1ex}{$r^2$}\!\left/ \!\raisebox{-1ex}{$R$}\right.\right)dr\\ & =& \left(8\mathrm{\pi }\raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}^2$}\right.\right)\left(\frac{r^2}{2}-\frac{r^3}{3R}\right)\\ & =& \left(8\mathrm{\pi }\raisebox{1ex}{$\mathrm{A}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}^2$}\right.\right)\left(\frac{0.{02}^2{\mathrm{m}}^2}{2}-\frac{0.{02}^3{\mathrm{m}}^2}{3\left(0.03\mathrm{m}\right)}\right)\\ & =& 2.79\times {10}^{-3}\mathrm{A}\end{array}$…………………………….. (I)

The integral is limited to $r$. Hence, by taking $R=r$in equation (i), the magnetic field can be determined as follows:

role="math" localid="1663225198778" $\begin{array}{rcl}B& =& \frac{{\mu }_0{i}_d{r}_1}{2\pi {r_1}^2}\\ & =& \frac{{\mu }_0{i}_d}{2\pi r_1}\\ & =& \frac{\left(4\pi \times {10}^{-7}\text{T.m/A}\right)\times \left(2.79\times {10}^{-3}\text{A}\right)}{2\times \pi \times \left(0.02\text{m}\right)}\\ & =& 2.79\times {10}^{-8}\text{T}\\ & =& 27.9\text{nT}\end{array}$

Therefore, the magnitude of the magnetic field due to displacement current at a radial distance $R\text{= 2.00cm}$is $B=27.9\text{nT}$.

(b) Determining the magnetic field's magnitude due to displacement current at a radial distance R=5.00 cm.

For the given radial distance $r_2=0.05\mathrm{m}$,$r_2>R$, the displacement current can be given using the given data in equation (I) of part (a) as follows: (The maximum value of $r_2$ will be R.)

$$\begin{gathered} i_{d,enc}=8\pi \left(\frac{R^2}{2}-\frac{R^3}{3R}\right) \\ =8\pi \left(\frac{{0.03}^2{\text{m}}^2}{2}-\frac{{0.03}^3{\text{m}}^3}{3\times \left(0.03\text{m}\right)}\right) \\ =3.77\times {10}^{-3}\text{A} \end{gathered}$$

By considering the real current $i$ and displacement current $i_d$equal, the magnetic field can be determined using the above and the given values in equation (ii) as follows:

role="math" localid="1663225693944" $$\begin{gathered} B=\frac{\left(4\pi \times {10}^{-7}\text{T.m/A}\right)\times \left(3.77\times {10}^{-3}\text{A}\right)}{2\times \pi \times \left(\text{0.05 m}\right)} \\ =1.51\times {10}^{-8}\text{T} \\ =15.1\text{nT} \end{gathered}$$

Therefore, the magnitude of the magnetic field due to displacement current at a radial distance $R=5.00\mathrm{cm}$is $B=15.1\text{nT}$.