91影视

Consider the following figure,

The figure shows the sphere, r is the radius of sphere, the point charge inside the sphere.

a)

Write the expression for the average field due to point charge q at a point which is distance r.

${\mathit{E}}_{\mathbf{a}\mathbf{v}\mathbf{e}}\mathbf{=}\frac{\mathbf{1}}{{\displaystyle \frac{\mathbf{4}}{\mathbf{3}}}\mathbf{蟺}{\mathbf{R}}^{\mathbf{3}}}\mathbf{鈭�}\mathit{E}\mathit{d}\mathit{味}$ 鈥︹�� (1)

Here, R is the radius of the sphere.

Substitute $\frac{1}{4蟺{蔚}_0}\frac{q\hat{}}{r^2}r$for E in equation (1)

$$\begin{gathered} E_{ave}=\frac{1}{\left({\displaystyle \frac{4}{3}}蟺R^3\right)}鈭�\frac{1}{4蟺{蔚}_0}\frac{q\hat{}}{r^2}rd味 \\ =\frac{1}{\left({\displaystyle \frac{4}{3}}蟺R^3\right)}\frac{1}{4蟺{蔚}_0}鈭�\frac{q\hat{}}{r^2}rd味 \\ =\frac{q}{\left({\displaystyle \frac{4}{3}}蟺R^3\right)}\frac{1}{4蟺{蔚}_0}鈭�\frac{1_{\hat{}}}{r^2}rd味 \\ =蚁\frac{1}{4蟺{蔚}_0}鈭�\frac{1_{\hat{}}}{r^2}rd味\left[\boxed{\xcancel{}}\frac{q}{\left({\displaystyle \frac{4}{3}}蟺R^3\right)}=蚁\right] \end{gathered}$$

Solve as further,

$$\begin{gathered} E_{ave}=\frac{1}{4蟺{蔚}_0}鈭�蚁\frac{1_{\hat{}}}{r^2}rd味 \\ =\frac{1}{4蟺{蔚}_0}鈭�\frac{蚁\hat{}}{r^2}rd味 \\ =E_{蚁} \end{gathered}$$

Hence, $E_{ave}=E_P$is proved.

b)

Write the expression for the electric filed inside a uniformly charged sphere of the charge density $蚁$.

$E_{蚁}=\frac{1}{3{蔚}_0}\widehat{蚁}r$ 鈥︹��. (2)

Substitute $\frac{q}{\left({\displaystyle \frac{4}{3}}蟺R^3\right)}$for $蚁$in equation (2).

$$\begin{gathered} E_{蚁}=\frac{1}{3{蔚}_0\left({\displaystyle \frac{-q}{\left({\displaystyle \frac{4}{3}}蟺R^3\right)}}\right)r} \\ =-\frac{q}{4蟺{蔚}_0{R}^3}r \\ =-\frac{蚁}{4蟺{蔚}_0{R}^3} \end{gathered}$$

Thus, the expression for the electric filed due to dipole moment $R_{蚁}=-\frac{蚁}{4蟺{蔚}_0{R}^3}$.

c)

Let鈥檚 consider that, many charges are present inside the sphere.

Then,

Write the expression for sum of the individual average electric field.

$E_{ave}=\frac{-P}{4蟺{蔚}_0{R}^3}$

Therefore, the individual average electric field is $E_{ave}=\frac{-P}{4蟺{蔚}_0{R}^3}$.

d)

The charge q is placed outside the sphere at r.

Write the expression for the electric field due to charge.

$E_{ave}=\frac{1}{4蟺{蔚}_0}\left(\frac{{\displaystyle \frac{4}{3}}蟺R^3蚁}{r^2}\right)r$

Write the expression for field at P due to uniformly charged sphere.

$R_{蚁}=\frac{1}{4蟺{蔚}_0}\frac{-q}{r^2}r$

Hence, average electric filed outside the sphere, due to point charge is equal to electric field is same charge produced at the center of the sphere.