Electric Forces & Fields

Electric field - Disc, shell, sphere

Electric field - Disc, shell, sphere

**Q1.** A thin disc of radius $$b=2 a$$ has a concentric hole of radius $$a$$ in it (see figure). It carries uniform surface charge $$\sigma$$ on it. If the electric field on its axis at height $$h$$ $$(h \ll a )$$ from its centre is given as $$C h$$ then value of $$C$$ is

A. $$\cfrac{\sigma}{4 a \in_{0}}$$ B. $$\cfrac{\sigma}{8 a \in_{0}}$$

C. $$\cfrac{\sigma}{a \in_{0}}$$ D. $$\cfrac{\sigma}{2 a \in_{0}}$$

**Q2.** Charge is distributed within a sphere of radius $$R$$ with a volume charge density $$\rho(r)=\cfrac{ A }{r^{2}} e^{-2 r / \alpha}$$, where $$A$$ and $$a$$ are constants. If $$Q$$ is the total charge of this charge distribution, the radius $$R$$ is

A. $$a \log \left(1-\cfrac{ Q }{2 \pi a A }\right)$$

B. $$\cfrac{a}{2} \log \left(\cfrac{1}{1-\cfrac{ Q }{2 \pi a A }}\right)$$

C. $$a \log \left(\cfrac{1}{1-\cfrac{ Q }{2 \pi a A }}\right)$$

D. $$\cfrac{a}{2} \log \left(1-\cfrac{Q}{2 \pi a A}\right)$$

**Q3.** Let a total charge $$2 Q$$ be distributed in a sphere of radius $$R$$, with the charge density given by $$\rho(r)=k r$$, where $$r$$ is the distance from the centre. Two charges $$A$$ and $$B$$, of $$-Q$$ each, are placed on diametrically opposite points, at equal distance $$a$$ from the centre. If $$A$$ and $$B$$ do not experience any force, then

A. $$a=8^{-1 / 4} R$$ B. $$a=\cfrac{3 R}{2^{1 / 4}}$$

C. $$a=2^{-1 / 4} R$$ D. $$a=\cfrac{R}{\sqrt{3}}$$

**Q4.** Consider a sphere of radius $$R$$ which carries a uniform charge density $$\rho$$. If a sphere of radius $$\cfrac{R}{2}$$ is carved out of it as shown, the ratio $$\cfrac{\left|\vec{E}_{A}\right|}{\left|\vec{E}_{B}\right|}$$ of magnitude of electric field $$\overrightarrow{ E }_{ A }$$ and $$\overline{ E }_{3}$$ respectively, at points $$A$$ and $$B$$ due to the remaining portion is

A. $$\cfrac{21}{34}$$ B. $$\cfrac{18}{54}$$ C. $$\cfrac{17}{54}$$ D. $$\cfrac{18}{34}$$

**Q5.** Let a total charge $$2 Q$$ be distributed in a sphere of radius $$R$$ with the charge density given by $$\rho(r)=k r$$, where $$r$$ is the distance from the centre. Two charges $$A$$ and $$B$$, of $$-Q$$ each, are placed on diametrically opposite points, at equal distance from the centre. If $$A$$ and $$B$$ do not experience any force, then

A. $$a=\cfrac{3 R}{2^{1 / 4}}$$ B. $$a=R / \sqrt{3}$$

C. $$a=8^{-1 / 4} R$$ D. $$a=2^{-1 / 4} R$$

**Q6.** In a uniformly charged sphere of total charge $$Q$$ and radius $$R$$, the electric field $$E$$ is plotted as a function of distance from the centre. The graph which would correspond to the above will be

**Q7.** Let there be a spherically symmetric charge distribution with charge density varying as $$\rho(r)=\rho_{0}\left(\cfrac{5}{4}-\cfrac{r}{R}\right)$$ upto $$r=R$$, and $$\rho(r)=0$$ for $$r>R$$, where $$r$$ is the distance from the origin. The electric field at a distance $$r(r

A. $$\cfrac{\rho_{0} r}{3 \epsilon_{0}}\left(\cfrac{5}{4}-\cfrac{r}{R}\right)$$

B. $$\cfrac{4 \pi \rho_{0} r}{3 \in_{0}}\left(\cfrac{5}{3}-\cfrac{r}{R}\right)$$

C. $$\cfrac{\rho_{0} r}{4 \in_{0}}\left(\cfrac{5}{3}-\cfrac{r}{R}\right)$$

D. $$\cfrac{4 \rho_{0} r}{3 \in_{0}}\left(\cfrac{5}{4}-\cfrac{r}{R}\right)$$

**Q8.** Let $$\rho(r)=\cfrac{Q}{\pi R^{4}} r$$ be the charge density distribution for a solid sphere of radius $$R$$ and total charge $$Q .$$ For a point $$p$$ inside the sphere at distance $$r_{1}$$ from the centre of the sphere, the magnitude of electric field is

A. $$\cfrac{Q r_{1}^{2}}{4 \pi \epsilon_{0} R^{4}}$$ B. $$\cfrac{Q r_{1}^{2}}{3 \pi \in_{0} R^{4}}$$

C. 0 D. $$\cfrac{Q}{4 \pi \in_{0} r_{1}^{2}}$$

**Q9.** A thin spherical shell of radius $$R$$ has a charge $$Q$$ spread uniformly over its surface. Which of the following graphs most closely represents the electric field $$E(r)$$ produced by the shell in the range $$0 \leq r<\infty$$, where $$r$$ is the distance from the centre of the shell

**Q10.** Two spherical conductors $$B$$ and $$C$$ having equal radii and carrying equal charges in them, repel each other with a force $$F$$ when kept apart at some distance.

A third spherical conductor having same radius as that of $$B$$ but uncharged, is brought in contact with $$B$$, then brought in contact with $$C$$ and finally removed away from both. The new force of repulsion between $$B$$ and $$C$$ is

A. $$\cfrac{F}{4}$$ B. $$\cfrac{3 F}{4}$$ C. $$\cfrac{F}{8}$$ D. $$\cfrac{3 F}{8}$$

**Q11.** A point charge $$Q_{1}=-125\ \mu C$$ is fixed at the center of an insulated disc of mass $$1\ kg$$. The disc rests on a rough horizontal plane. Another charge $$Q_{2}$$ $$=125\ \mu C$$ is fixed vertically above the center of the disc at a height $$h=1$$
m.

After the disc is displaced slightly in the horizontal direction (friction is sufficient to prevent slipping), find the period of oscillation of disc.

**Q12.** The electric field intensity at the center of a uniformly charged hemispherical shell is $$E_{0}$$. Now two portions of the hemisphere are cut from either side, and the remaining portion is shown in figure. If $$\alpha=\beta=\pi / 3$$, then the electric field intensity at the center due to the remaining portion is

A. $$E_{0} / 3$$ B. $$E_{0} / 6$$

C. $$E_{0} / 2$$ D. Information insufficient

**Q13.** A non-conducting sphere of radius $$R$$ is filled with uniform volume charge density $$-\rho .$$ The center of this sphere is displaced from the origin by $$d$$. The electric field $$\vec{E}$$ at any point $$P$$ having position vector $$\vec{r}$$ inside the sphere is

A. $$\cfrac{\rho}{3 \varepsilon_{0}} \vec{d}$$ B. $$\cfrac{\rho}{3 \varepsilon_{0}}(\vec{r}-\vec{d})$$

C. $$\cfrac{\rho}{3 \varepsilon_{0}}(\vec{d}-\vec{r})$$ D. $$\cfrac{\rho}{3 \varepsilon_{0}}(\vec{d}+\vec{r})$$

**Q14.** A tiny sphere has charge $$Q$$ on it. A portion of charge $$x$$ is transferred to a second nearby sphere. spheres are fixed and small enough to be treated as particles. When the ratio $$\cfrac{x}{Q}$$ is equal to $$\eta_{0}$$, the force
between the two spheres is found to be maximum with a value equal to $$F_{0}$$.

A. Find $$\eta_{0}$$.

B. Find the ratio $$\cfrac{x}{Q}$$ for which the force between the spheres is $$75 \%$$ of $$F_{0}$$.

**Q15.** A uniformly charged disc lies in $$x y$$ plane with its centre at the origin. The strength of electric field (E) changes along the $$z$$ axis according to the graph shown. Find the radius of the disC.

**Q16.** A hollow dielectric sphere, as shown in the figure, has inner and outer radii of $$R_{1}$$ and $$R_{2}$$ respectively. The total charge sphere is $$+Q$$. This charge is uniformly distributed in the volume of the dielectriC. $$A$$ point charge $$Q_{0}$$ is placed at the centre and it is seen that the electric field in the region $$R_{1}

A. The value of $$Q_{0}$$ is $$\cfrac{Q R_{1}^{3}}{R_{2}^{3}-R_{1}^{3}}$$.

B. The electric field for $$r>R_{2}$$ is given by $$\cfrac{Q+Q}{4 \pi \varepsilon_{0} r^{2}}$$.

C. The electric field for $$r

D. None of the above are true.

**Q17.** A positive charge $$q$$ is placed in a spherical cavity made in a positively charged sphere. The centres of sphere and cavity are displaced by a small distance $$\vec{l}$$. Force on charge $$q$$ is:

A. in the direction parallel to vector $$\vec{l}$$

B. in radial direction

C. in a direction which depends on the magnitude of charge density in sphere

D. direction can not be determineD.

**Q18.** Two spherical, nonconducting. and very thin shells having uniformly distributed positive charge $$Q$$ on their surfaces are located a distance $$10 d$$ from each other. Radius of each shell is $$d$$.

A positive point charge $$q$$ is placed inside one of the shells at a distance $$d / 2$$ from the center, on the line connecting the centers of the two shells. What is the net force on the charge $$q$$ ?

A. $$\cfrac{q Q}{361 \pi \varepsilon_{0} d^{2}}$$ to the left

B. $$\cfrac{q Q}{361 \pi \varepsilon_{0} d^{2}}$$ to the right

C. $$\cfrac{362 q Q}{361 \pi \varepsilon_{0} d^{2}}$$ to the left

D. $$\cfrac{360 q Q}{361 \pi \varepsilon_{0} d^{2}}$$ to the right

**Q19.** Volume charge density inside a sphere of radius $$R$$ is proportional to the distance $$(x)$$ from the centre and is given by $$\rho=\alpha x$$, where $$\alpha$$ is a constant. The strength of electric field at a distance $$R / 2$$ from the centre is

A. $$\cfrac{\alpha R^{3}}{4 \varepsilon_{0}}$$ B. $$\cfrac{\alpha R^{2}}{4 \varepsilon_{0}}$$

C. $$\cfrac{\alpha R^{2}}{3 \varepsilon_{0}}$$ D. none of these

**Q20.** A solid sphere of radius $$R$$ has a charge $$Q$$ distributed in its volume with a charge density $$\rho=k r^{a}$$, where $$k$$ and $$a$$ are its volume constants and $$r$$ is the distance from its centre. If the electric field at $$r=R / 2$$ is $$1 / 8$$ times that at $$r=R$$, find the value of $$a$$.

**Q21.** The diagram shows a uniformly charged hemisphere of radius $$R$$. It has volume charge density $$\rho$$. If the magnitude of electric field at a point $$A$$ located a distance $$2 R$$ above its centre is $$E$$ then what is the electric field at the point $$B$$ which is $$2 R$$ below its centre as shown in figure.

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