Electric Forces & Fields
Electric field - Disc, shell, sphere
DPP # EFF-05

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. 