91Ó°ÊÓ

Write the expression for the angular momentum of the fields.

$\mathit{L}\mathbf{=}\mathbf{∫}\left(r×g\right)\mathit{d}\mathit{τ}$ …… (1)

Here, g is the momentum density.

(a)

Write the expression for the momentum density.

$g=\mathrm{Î\mu }\left(E×B\right)$ …… (2)

Here, E is the electric field and B is the magnetic field.

Write the expression for the electric field.

$E=\frac{Q}{4\mathrm{Ï€}\mathrm{Î\mu }}\frac{1}{r^2}\hat{r}$

Substitute the values in equation (2).

localid="1657537044442" $$\begin{gathered} g={\mathrm{Î\mu }}_0\left(\frac{Q}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{1}{r^2}\hat{r}×B\right) \\ g=\frac{Q{B}_0}{4\mathrm{Ï€}{r}^2}\left(\hat{r}×\hat{z}\right) \end{gathered}$$

Substitute the known values in equation (1).

$$\begin{gathered} L=∫\left(r×\frac{QB}{4\mathrm{π}{r}^2}\left(\hat{r}×\hat{z}\right)\right)d\mathrm{τ} \\ L=\frac{Q{B}_0}{4\mathrm{π}r}∫\frac{1}{r^2}\left[r×\left(\hat{r}×\hat{z}\right)\right]r^2\sin \mathrm{θ}drd\mathrm{θ}d \\ L=\frac{Q{B}_0}{4\mathrm{π}r}∫r\left[\hat{r}\left[\hat{r}×\hat{z}\right]\right]\sin \mathrm{θ}drd\mathrm{θ}d......\left(3\right) \end{gathered}$$

Since,

$$\begin{gathered} \hat{r}×\left(\hat{r}×\hat{z}\right)=\hat{r}\left(\hat{r}.\hat{z}\right)-\hat{z}\left(\hat{r}.\hat{r}\right) \\ \hat{r}×\left(\hat{r}×\hat{z}\right)=\hat{r}\cos \mathrm{θ} \\ \hat{r}×\left(\hat{r}×\hat{z}\right)=\hat{z} \end{gathered}$$

As L has to be along the z-direction, pick the z component of $\hat{r}$. Hence,

localid="1657537019616" $$\begin{gathered} {\left[\hat{r}×\left(\hat{r}×\hat{z}\right)\right]}_z={\cos }^2\mathrm{θ}-1 \\ \left[\hat{r}×{\left(\hat{r}×\hat{z}\right)}_z\right]=-{\sin }^2\mathrm{θ} \end{gathered}$$

From equation (3),

localid="1657534367950" $$\begin{gathered} L=-\frac{Q{B}_0}{4\mathrm{π}r}∫r{\sin }^3\mathrm{θ}drd\mathrm{θ}d\mathrm{ϕ} \\ L=-\frac{Q{B}_0}{4\mathrm{π}r}\left(2\mathrm{π}\right){∫}_0^{\mathrm{π}}{\sin }^3\mathrm{θ}d\mathrm{θ}{∫}_a^{b}rdr \\ L=-\frac{Q{B}_0}{2}\left(\frac{4}{3}\right)\left(\frac{b^2-a^2}{2}\right) \\ L=-\frac{1}{3}Q{B}_0\left(b^2-a^2\right)\hat{z} \end{gathered}$$

Therefore, the angular momentum of the fields is $L=-\frac{1}{3}Q{B}_0\left(b^2-a^2\right)\hat{z}$.

(b)

Write the expression for the torque on the patch.

$N=s×dF$ …… (4)

Here, dF is the force on a patch.

Write the expression for the force on a patch.

$dF=E\mathrm{σ}da$ …… (5)

Write the expression for the electric field.

$E=-\frac{s}{2}\frac{dB}{dt}\widehat{\mathrm{Ï•}}$

Substitute the known values in equation (5).

$$\begin{gathered} dF=\left(-\frac{s}{2}\frac{dB}{dt}\widehat{\mathrm{ϕ}}\right)\mathrm{σ}da\left(\begin{array}{c}\begin{array}{c}∴s=a\sin \mathrm{θ}\hat{s}\\ da=a^2\sin \mathrm{θ}d\mathrm{θ}d\mathrm{ϕ}\end{array}\end{array}\right) \\ dF=\left(-\frac{s}{2}\frac{dB}{dt}\widehat{\mathrm{ϕ}}\frac{dB}{dt}\widehat{\mathrm{ϕ}}\right)\mathrm{σ}\left(a^2\sin \mathrm{θ}d\mathrm{θ}d\mathrm{ϕ}\right) \end{gathered}$$

Substitute the known values in equation (4) to calculate the net torque on the sphere at radius a.

$$\begin{gathered} N=s×\left(-\frac{a\sin \mathrm{θ}\hat{s}}{2}\frac{dB}{dt}\widehat{\mathrm{ϕ}}\right)\mathrm{σ}\left(a^2\sin \mathrm{θ}d\mathrm{θ}d\mathrm{ϕ}\right) \\ {N}_a=--\frac{(a\sin \mathrm{θ})}{2}\frac{dB}{dt}\mathrm{σ}{a}^3{\sin }^2\mathrm{θ}\left(\hat{s}×\widehat{\mathrm{ϕ}}\right)\left(d\mathrm{θ}d\mathrm{ϕ}\right) \\ {N}_a=--\frac{(a^4{\sin }^3\mathrm{θ})}{2}\frac{dB}{dt}\left(\frac{Q}{4\mathrm{π}{a}^2}\right)\left(\hat{s}×\widehat{\mathrm{ϕ}}\right)\left(d\mathrm{θ}d\mathrm{ϕ}\right) \\ {N}_a=-\frac{Q{a}^2}{8\mathrm{π}}\frac{dB}{dt}\hat{z}\left(2\mathrm{π}\right){∫}_0^{\mathrm{π}}{\sin }^3\mathrm{θ}d\mathrm{θ}{∫}_0^{2\mathrm{π}}d\mathrm{ϕ} \end{gathered}$$

On further solving,

$$\begin{gathered} N_a=-\frac{Q{a}^2}{8}\frac{dB}{dt}\left(\hat{z}\right)\left(2\mathrm{Ï€}\right)\left(\frac{4}{3}\right) \\ {N}_a=-\frac{Q{a}^2}{3}\frac{dB}{dt}\hat{z} \end{gathered}$$

Calculate the net torque on the sphere at radius b.

$N_b=\frac{Q{b}^2}{3}\frac{dB}{dt}\hat{z}$

Hence, the total torque on each sphere will be,

localid="1657536473852" $$\begin{gathered} N=N_b-N_a \\ N=\left(\frac{Q{b}^2}{3}\frac{dB}{dt}\hat{z}\right)-\left(\frac{Q{a}^2}{3}\frac{dB}{dt}\hat{z}\right) \\ N=\frac{Q}{3}\frac{dB}{dt}\left(b^2-a^2\right)\hat{z} \end{gathered}$$

Calculate the angular momentum delivered to the spheres.

$$\begin{gathered} L=∫Ndt \\ L=∫\frac{Q}{3}\frac{dB}{dt}\left(b^2-a^2\right)\hat{z}dt \\ L=\frac{Q}{3}\left(b^2-a^2\right)\hat{z}{∫}_{B_0}^0 \\ L=-\frac{1}{3}Q{B}_0\left(b^2-a^2\right)\hat{z} \end{gathered}$$

Therefore, the net torque is $N=\frac{Q}{3}\frac{dB}{dt}\left(b^2-a^2\right)\hat{z}$and the angular momentum is $L=-\frac{1}{3}Q{B}_0\left(b^2-a^2\right)\hat{z}$.