Question

A solid plastic sphere of radius ${\mathrm{R}}_1$has a charge $-{\mathrm{Q}}_1$on its surface (Figure 15.70). A concentric spherical metal shell of inner radius ${\mathrm{R}}_2$and outer radius ${\mathrm{R}}_3$carries a charge ${\mathrm{Q}}_2$on the inner surface and a charge ${\mathrm{Q}}_3$on the outer surface. ${\mathrm{Q}}_1$, ${\mathrm{Q}}_2$, and ${\mathrm{Q}}_3$are positive numbers, and the total charge ${\mathrm{Q}}_2$$+{\mathrm{Q}}_3$on the metal shell is greater than ${\mathrm{Q}}_1$.

At an observation location a distance $\mathrm{r}$from the center, determine the magnitude and direction of the electric field in the following regions, and explain briefly in each case. For parts role="math" localid="1656931802199" $\left(\mathrm{a}\right)-\left(\mathrm{d}\right)$, be sure to give both the direction and the magnitude of the electric field, and explain briefly: (a)role="math" localid="1656932347681" $r<R_1$(inside the plastic sphere), (b)role="math" localid="1656932286893" $R_1<r<R_2$(in the air gap), (c)role="math" localid="1656932322994" $R<r<R$(in the metal),(d)role="math" localid="1656932390135" $\mathrm{r}>{\mathrm{R}}_3$(outside the metal).(e) Supposerole="math" localid="1656932377163" $-{\mathrm{Q}}_1=-5n\mathrm{C}$. What isrole="math" localid="1656932400004" ${\mathrm{Q}}_2$? Explain fully on the basis of fundamental principles. (f) What can you say about the molecular polarization in the plastic? Explain briefly. Include a drawing if appropriate.

Short answer

(a) The electric field inside the plastic sphere is 0.

(b) The electric field in the air gap is $\frac{-Q_1}{{\epsilon }_{0}4\pi r^2}$and its direction is negative.

(c) The electric field in the metal is 0.

(d) The electric field outside the metal is$\frac{Q_3}{{\epsilon }_{0}4\pi r^2}$and it is directed outwards from the sphere’s center.

(e) The value of the charge $Q_2$is$5\mathrm{nC}$ .

(f) The plastic’s dipole’s negative part will be outward and the positive part will be towards the center.

Step-by-step solution

Identification of given data

The given data can be listed below as:

• The radius of the solid plastic sphere is,$R_1$
• The charge of the solid plastic sphere is,$-Q_1$$-Q_1$
• The inner radius of the concentric spherical metal shell is,$R_2$
• The inner surface charge of the concentric spherical metal shell is,$Q_2$
• The outer radius of the concentric spherical metal shell is,$R_3$
• The outer surface charge of the concentric spherical metal shell is,$Q_3$

Significance of the electric flux

The electric flux is described as the product of the electric field and the special surface’s area. Moreover, the electric flux is also referred to as the division of the charge of an object with the permittivity.

(a) Determination of the magnitude and direction of the electric field inside the plastic sphere

The gaussian surface at the point $r<R_1$will take no charge. However, by the symmetry of the Gaussian surface, the electric field is equal to zero at all the points inside the plastic sphere.

Thus, the electric field inside the plastic sphere is 0.

(b) Determination of the magnitude and direction of the electric field in the air gap

A Gaussian surface as a sphere has been considered at the radius $R_1<r<R_2$. However, the magnitude of the electric field is the same at all the points on the surface, and also it is directed towards the center.

The equation of the electric flux is expressed as:

$\varphi =EA$

…(i)

Here, $E$is the electric field and $A$is the spherical surface’s area.

For the charge $-Q_1$, the equation of the electric flux is expressed as:

$\varphi =\frac{-Q_1}{{\epsilon }_0}$

…(ii)

Here, is the charge of the solid plastic sphere and is the permittivity.

Equating equation (i) and (ii),

$EA=\frac{-Q_1}{{\epsilon }_0}$

Substitute for in the above equation.

role="math" localid="1656933780132" $$\begin{gathered} E.4\pi r^2=\frac{-Q_1}{{\epsilon }_0} \\ E=\frac{-Q_1}{{\epsilon }_{0}4\pi r^2} \end{gathered}$$

Thus, the electric field in the air gap and its direction is negative.

(c) Determination of the magnitude and direction of the electric field in the metal

As inside the metal, the value of the inner surface charge of the concentric spherical metal shell and charge of the solid plastic sphere is equal, that is $Q_1=Q_2$. Hence, at this point $R_2<r<R_3$, the charge inside any type of spherical surface is zero.

Thus, the electric field in the metal is $0$.

(d) Determination of the magnitude and direction of the electric field outside the metal

For $r>R_3$,

For the charge $Q_1$, $Q_2$and $Q_3$, the equation of the electric flux is expressed as:

$\varphi ={\frac{-Q_1+Q_2+Q_3}{{\epsilon }_0}}^{}$

…(iii)

Here, is the charge of the solid plastic sphere, the inner and the outer surface charge of the concentric spherical metal shell are $Q_2$and $Q_3$and ${\epsilon }_0$is the permittivity.

Equating the equation (i) and (iii),

$EA=\frac{-Q_1+Q_2+Q_3}{{\epsilon }_0}$

Substitute $4\pi r^2$for $A$and $Q_1$for $Q_2$in the above equation.

$E·4\pi r^2=\frac{-Q_1+Q_1+Q_3}{{\epsilon }_0}$

$E·4\pi r^2=\frac{Q_3}{{\epsilon }_0}$

$E=\frac{Q_3}{{\epsilon }_{0}4\pi r^2}$

Thus, the electric field outside the metal is $\frac{Q_3}{{\epsilon }_{0}4\pi r^2}$and it is directed outwards from the sphere’s center.

(e) Determination of the charge Q2

Here the value of the charge $-Q_1$is given as $-5\mathrm{nC}$. As the electric field inside the particular conductor is zero, then for the radius $R_2<r<R_3$, the net charge inside the sphere is zero. Hence, the value of the second charge becomes $Q_1=Q_2$.

Thus, the value of the charge $Q_2$is $5\mathrm{nC}$.

(f) Determination of the molecular polarization in the plastic

The molecular polarization is described as the ratio of the dipole moment to the local electric field. However, inside a particular crystalline solid, the dipole moment is being considered per unit cell.

Thus, the plastic’s dipole’s negative part will be outward and the positive part will be towards the center.