Electric Forces & Fields
Electric field - Line, Sheet, Cylinder
DPP # EFF-06

Q1. In the given arrangement of a charged square frame made up of four wires $$1,\ 2,\ 3$$ and $$4$$ charged with the linear charge density as mentioned in figure. Find electric field at centre due to this frame. Q2. A system consists of a thin charged wire ring of radius $$R$$ and a very long uniformly charged thread oriented along the axis of the ring, with one of its ends coinciding with the centre of the ring. The total charge of the ring is equal to $$q$$. The charge of the thread (per unit length) is equal to $$\lambda$$. Find the interaction force between the ring and the thread.

Q3. In the given arrangement find the electric field at $$C$$ in the figure. Here, the U-shaped wire is uniformly charged with linear charge density $$\lambda$$. Q4. In the given arrangement, find electric field at $$C$$. Complete wire is uniformly charged at linear charge density $$\lambda$$. Q5. Given an equilateral triangle mode up of three rods each of length $$l$$. Find electric field strength at the centroid of triangle. The linear charge density on the sides of triangle are as shown in figure. Q6. A positive charge $$q$$ is placed in front of a conducting solid cube at a distance $$d$$ from its centre. Find the electric field at the centre of the cube due to the charges appearing on its surface.

Q7. A large nonconducting surface has a uniform charge density $$\sigma$$. A small circular hole of radius $$R$$ is cut in the middle of the sheet, as shown in figure. Ignore fringing of the field lines around all edges, calculate the electric field at point $$P$$ at a distance $$z$$ from the centre of the hole along its axis. Q8. An infinitely long cylindrical shell of inner radius $$r_{1}$$ and outer radius $$r_{2}$$ is charged in its volume with a volume charge density which varies with distance from axis of cylinder as $$\rho=b / r\ C / m^{3}$$ where $$C$$ is a positive constant and $$r$$ is the distance from axis of cylinder. Find the electric field intensity at a point $$P$$ at a distance $$x$$ from axis of cylinder. Q9. Three infinite long charged sheets of charge densities $$-\sigma,-2 \sigma$$ and $$\sigma$$ are placed parallel to $$x y$$ -plane at $$z=0,\ z=a,\ z=3 a$$. Electric field at point $$P$$ is given as A. $$-\cfrac{2 \sigma}{\epsilon_{0}} \hat{k}$$       B. $$\cfrac{2 \sigma}{\epsilon_{0}} \hat{k}$$

C. $$\cfrac{-4 \sigma}{\epsilon_{0}} \hat{k}$$       D. $$\cfrac{4 \sigma}{\epsilon_{0}} \hat{k}$$

Q10. A particle of charge $$-q$$ and mass $$m$$ moves in a circle of radius $$r$$ around an infinitely long line charge of linear charge density $$+\lambda$$. Then the time period of revolution of the particle will be $$\left(k=\cfrac{1}{4 \pi \varepsilon_{0}}\right)$$

A. $$T=2 \pi r \sqrt{\cfrac{m}{2 k \lambda q}}$$

B. $$T^{2}=\cfrac{4 \pi^{2} m}{2 k \lambda q} r^{3}$$

C. $$T=\cfrac{1}{2 \pi r} \sqrt{\cfrac{2 k \lambda q}{m}}$$

D. $$T=\cfrac{1}{2 \pi r} \sqrt{\cfrac{m}{2 k \lambda q}}$$

Q11. An infinitely long wire having uniform linear charge density $$\cfrac{10}{9}\ nC / m$$ is kept along z-axis from $$z=-\infty$$ to $$z=+\infty$$. The electric field $${E}$$ at point $$(6\ cm,\ 8\ cm,\ 10 \ cm )$$ will be A. $$(160 i+120)+200 \hat{k}) N / C$$

B. $$(200 \hat{k}) N / C$$

C. $$(160 \hat{i}+120 \hat{\jmath}) N / C$$

D. $$(120 \hat{t}+160 \hat{\jmath}) N / C$$

Q12. Two semicircular rings lying in same plane and of uniform linear charge density $$\lambda$$ have radius $$r$$ and $$2 r$$. They are joined using two straight uniformly charged wires of linear charge density $$\lambda$$ and length $$r$$ as shown in the figure. The magnitude of electric field at common centre of semi circular rings is A. $$\cfrac{1}{4 \pi e_{0}} \cfrac{32}{2 r}$$

B. $$\cfrac{1}{4 \pi e,} \cfrac{3}{2 r}$$

C. $$\cfrac{1}{4 \pi e_{9}} \cfrac{21}{r}$$

D. $$\cfrac{1}{4 \pi e_{y}} \cfrac{1}{r}$$

Q13. Two mutually perpendicular wires carry charge densities $$\lambda_{1}$$ and $$\lambda_{2} .$$ If the electric line of force makes an angle $$\alpha$$ with the second wire, then find $$\cfrac{\lambda_{1}}{\lambda_{2}}$$ in terms of angle $$\alpha$$. A. $$\cot ^{2} \alpha$$   B. $$\tan ^{2} \alpha$$   C. $$\cot \alpha$$   D. $$\tan \alpha$$

Q14. A positive point charge is released from rest at a distance $$r_{0}$$ from a positive line charge with uniform density. The speed $$(v)$$ of the point charge, as a function of instantaneous distance $$r$$ from line charge, is proportional to A. $$v \propto e^{+r / r_{0}}$$       B. $$v \propto \ln \left(\cfrac{r}{r_{0}}\right)$$

C. $$v \propto\left(\cfrac{r}{r_{0}}\right)$$       D. $$v \propto \sqrt{\ln \left(\cfrac{r}{r_{0}}\right)}$$

Q15. Two identical large thin conducting plates are placed parallel to each other as shown in figure. They are carrying charges $$Q^{\prime}$$ and $$3 Q$$ ' respectively. The variation of electric field as a function at $$x$$ (for $$x=0$$ to $$x=3 d$$ ) will be best represented by. A. B. C. D. Q16. An infinitely large non-conducting plane has uniform surface charge density $$\sigma .$$ It has a circular hole carved out from it. The electric field at a point which is at a distance ' $$a$$ ' from the centre of the hole (perpendicular to the plane) is $$\cfrac{\sigma}{2 \sqrt{2} \epsilon_{0}}$$. The radius of the hole is

A. $$2 a$$   B. $$a$$   C. $$3 a$$   D. none of these.

Q17. A large sheet carries uniform surface charge density $$\sigma$$. A non conducting rod of length 2 / has a linear charge density $$\lambda$$ on one half and $$-\lambda$$ on the second half. The rod is hinged at mid point and makes an angle $$\theta$$ with the normal to the sheet. The torque experienced by the rod is A. $$0$$       B. $$\cfrac{\sigma \lambda l^{2}}{2 \varepsilon_{0}} \sin \theta$$

C. $$\cfrac{\sigma \lambda l^{2}}{\varepsilon_{0}} \sin \theta$$       D. $$\cfrac{\sigma \lambda l}{2 \varepsilon_{0}}$$

Q18. Two large thin conducting plates with small gap in between are placed in a uniform electric field $$E$$ (perpendicular to the plates). Area of each plate is $$A$$ and charges $$+Q$$ and $$-Q$$ are given to these plates as shown in the figure. $$R, S$$ and $$T$$ (see figure) are three points in space, then the A. field at point $$R$$ is $$E$$

B. charge on the inner wall of the left plate will be $$\left(Q+\varepsilon_{0} A E\right)$$

C. field at point $$T$$ is $$\left(E+\cfrac{Q}{\varepsilon_{0} A}\right)$$

D. field at point $$S$$ is $$\left(E+\cfrac{Q}{A \varepsilon_{0}}\right)$$

Q19. Two infinitely long line charges, having charge density $$\lambda$$ each, are parallel to each other and separated by a distance $$d$$. A particle having mass $$m$$ and charge $$q$$ is placed symmetrically between them such that the two lines and the charge lie in the same plane.

This charge is displaced slightly in the plane along the line $$A B$$, which is perpendicular to the line charges, and released. Neglect gravity. A. The motion of the particle will be SHM if $$\lambda q>0 .$$

B. The motion of the particle will be SHM if $$\lambda q<0 .$$

C. In case the motion is SHM the time period 1 directly proportional to $$d$$.

D. None of the above.

Q20. A uniform nonconducting rod of mass $$m$$ and length $$\ell$$, with charge density $$\lambda$$, as shown in figure, is hinged at the midpoint at origin so that it can rotate in a horizontal plane without any friction. A uniform electric field $$E$$ exists parallel to $$x$$ -axis in the entire region. Calculate the period of small oscillations of the rod. Q21. Find the electric field vector at $$P(a, a, a)$$ due to three infinitely long lines of charges along the $$x-, y$$ - and $$z$$ -axes, respectively. The charge density, i.e., charge per unit length of each wire is $$\lambda$$. A. $$\cfrac{\lambda}{3 \pi \varepsilon_{0} a}(\hat{i}+\hat{j}+\hat{k})$$

B. $$\cfrac{\lambda}{2 \pi \varepsilon_{0} a}(\hat{i}+\hat{j}+\hat{k})$$

C. $$\cfrac{\lambda}{2 \sqrt{2} \pi \varepsilon_{0} a}(\hat{i}+\hat{j}+\hat{k})$$

D. $$\cfrac{\sqrt{2} \lambda}{\pi \varepsilon_{0} a}(\hat{i}+\hat{j}+\hat{k})$$

Q22. Find the force experienced by a semicircular rod having a charge $$q$$ as shown in the figure. Radius of the wire is $$R$$, and the line of charge with linear charge density $$\lambda$$ passes through its center and is perpendicular to the plane of wire. A. $$\cfrac{\lambda q}{2 \pi^{2} \varepsilon_{0} R}$$       B. $$\cfrac{\lambda q}{\pi^{2} \varepsilon_{0} R}$$

C. $$\cfrac{\lambda q}{4 \pi^{2} \varepsilon_{0} R}$$       D. $$\cfrac{\lambda q}{4 \pi \varepsilon_{0} R}$$

Q23. The direction $$(\theta)$$ of $$\vec{E}$$ at point $$P$$ due to uniformly charged finite rod will be A. at an angle $$30^{\circ}$$ from the $$x$$ -axis

B. $$45^{\circ}$$ from $$x$$ -axis

C. $$60^{\circ}$$ from $$x$$ -axis

D. none of these

Q24. A large sheet carries uniform surface charge density $$\sigma$$. A rod of length $$2 l$$ has a linear charge density $$\lambda$$ on one half and $$-\lambda$$ on the second half. The rod is hinged at the midpoint $$O$$ and makes an angle $$\theta$$ with the normal to the sheet. The torque experienced by the rod is A. 0       B. $$\cfrac{\sigma \lambda l^{2}}{2 \varepsilon_{0}} \sin \theta$$

C. $$\cfrac{\sigma \lambda l^{2}}{\varepsilon_{0}} \sin \theta$$       D. $$\cfrac{\sigma \lambda l}{2 \varepsilon_{0}}$$