Question

The figure shows a circular region of radius$\mathbf{R}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{cm}$in which a uniform displacement current ${\mathit{i}}_{\mathbf{d}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{500}\mathbf{}\mathit{A}$isout of the page.(a)What is the magnitude of the magnetic field due to displacement current at a radial distance$\mathit{r}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{}\mathit{c}\mathit{m}$?(b)What is the magnitude of the magnetic field due to displacement current at a radial distancerole="math" localid="1663231983050" $\mathbf{r}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{cm}$?

Short answer

The magnitude of the magnetic field due to displacement current at the radial distance

(a) $B=2.22\mu T$

(b)role="math" localid="1663230482655" $B=2.00\mu T$

Step-by-step solution

Given data

Displacement current,$i_d=0.500\text{A}$

Radius of circular region,$R=3.00cm=3.00\times {10}^{-2}m$

Concept

For the uniform displacement current density, determine the magnitude of the magnetic field for the circular region of radius Ror the cases: By using the relation between the current density and the displacement current, and the expression of flux in terms of area and electric field, find the current density as a function of the electric field. In electromagnetism, displacement current density is the quantity appearing in Maxwell equations that are defined in terms of the rate of change of D, the electric displacement field.

$$\begin{gathered} \mathbf{i}\mathbf{)}\mathbf{}\mathbf{r}\mathbf{<}\mathbf{R} \\ \mathbf{ii}\mathbf{)}\mathbf{}\mathbf{r}\mathbf{>}\mathbf{R} \end{gathered}$$

where is the radius of the Amperian loop.

Formulae are as follows:

For,$r<R$

$B=\frac{{\mu }_0{i}_{d}r}{2\pi R^2}$

For,$r>R$

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

where, B is the magnetic field, i is current, and Ris a radial distance.

(a) Determining themagnitude of the magnetic field due to displacement current at a radial distance

For the given circular region, the magnitude of the magnetic field due to displacement current is given as follows,

The area of the circular plate is,

For $r=2.00cm$ , $r<R$

$B=\frac{{\mu }_0{i}_{d}r}{2\pi R^2}$

role="math" localid="1663231209798" $$\begin{gathered} B=\frac{4\times \pi \times {10}^{-7}\times 0.50\times 0.02}{2\times \pi \times {0.03}^2} \\ B=2.22\times {10}^{-6}\text{T} \\ \text{= 2.22} \mu \text{T} \end{gathered}$$

Therefore, the magnitude of the magnetic field due to displacement currentat a radial distance role="math" localid="1663231286057" $r=2.00cm$is $B=2.22\mu T$.

(b) Determining the magnitude of the magnetic field due to displacement current at a radial distance

For $r=5.00cm$,$r>R$

$$\begin{gathered} B=\frac{{\mu }_0{i}_d}{2\pi r} \\ =\frac{4\times \pi \times {10}^{-7}\times 0.50}{2\times \pi \times 0.05} \\ =2.00\times {10}^{-6}\text{T} \\ =2.00 \mu T \end{gathered}$$

Therefore, themagnitude of the magnetic field due to displacement currentat a radial distance localid="1663232028142" $r=5.00cm$is $B=2.00\mu T$.

The magnitude of the magnetic field is due to the displacement current using the displacement current density, when the Amperian loop is smaller and larger than the given circular area.