Electric Forces & Fields

Electric field due to Ring and Arc

Electric field due to Ring and Arc

**Q1.** For a uniformly charged ring of radius $$R$$, the electric field on its axis has the largest magnitude at a distance $$h$$ from its centre. Then value of $$h$$ is

A. $$\cfrac{ R }{\sqrt{5}}$$ B. $$\cfrac{R}{\sqrt{2}}$$ C. $$R$$ D. $$R \sqrt{2}$$

**Q2.** Two concentric rings, one of radius $$R$$ and total charge $$+Q$$ and the second of radius $$2 R$$ and total charge $$Q$$, lie in $$x$$ - $$y$$ plane (i.e., $$z=0$$ plane). The common centre of rings lies at origin and the common axis coincides with $$z$$ -axis. The charge is uniformly distributed on both rings. At what distance from origin is the net electric field on $$z$$ -axis zero?

A. $$\cfrac{R}{2}$$ B. $$\cfrac{R}{\sqrt{2}}$$ C. $$\cfrac{R}{2 \sqrt{2}}$$ D. $$\sqrt{2} R$$

**Q3.** Five point charges each having magnitude $$q$$ are placed at the corner of hexagon as shown in the figure. Net electric field at the centre $$O$$ is $$\vec{E}$$. To get net electric field at $$O$$ be $$6 \vec{E}$$, charge placed on the remaining sixth corner should be

A. $$6 q$$ B. $$-6 q$$ C. $$5 q$$ D. $$-5 q$$

**Q4.** A thin semi-circular ring of radius $$r$$ has a positive charge $$q$$ distributed uniformly over it. The net field $$\vec{E}$$ at the centre $$O$$ is

A. $$\cfrac{q}{2 \pi^{2} \varepsilon_{0} r^{2}} \hat{j}$$ B. $$\cfrac{q}{4 \pi^{2} \varepsilon_{0} r^{2}} \hat{j}$$

C. $$-\cfrac{q}{4 \pi^{2} \varepsilon_{0} r^{2}} \hat{j}$$ D. $$-\cfrac{q}{2 \pi^{2} \varepsilon_{0} r^{2}} \hat{j}$$

**Q5.** Two concentric rings, one of radius $$a$$ and the other of radius $$b$$ have the charges $$+q$$ and $$-\left(\frac{2}{5}\right)^{-\frac{3}{2}} q$$, respectively, as shown in the figure.

Find the ratio $$\cfrac{b}{a}$$, if a charge particle placed on the axis at $$z=a$$ is in equilibrium.

A. $$4: 1$$ B. $$2 \sqrt{2}: 1$$

C. $$2: 1$$ D. $$8: 1$$

**Q6.** The charge per unit length of the four quadrant of the ring is $$2 \lambda,\ -2 \lambda,\ \lambda$$ and $$-\lambda$$, respectively. What is the electric field at the centre?

A. $$\cfrac{-\lambda}{2 \pi \varepsilon_{0} R} \hat{i}$$ B. $$\cfrac{\lambda}{2 \pi \varepsilon_{0} R} \hat{j}$$

C. $$\cfrac{\sqrt{2} \lambda}{4 \pi \varepsilon_{0} R} \hat{i}$$ D. None of these

**Q7.** A rod of length $$L$$ has a total charge $$Q$$ distributed uniformly along its length. It is bent in the shape of a semicircle. Find the magnitude of the electric field at the centre of curvature of the semicircle.

A. $$\cfrac{Q}{2 \varepsilon_{0} L^{2}}$$ B. $$\cfrac{Q}{\varepsilon_{0} L^{2}}$$

C. $$\cfrac{2 Q}{\varepsilon_{0} L^{2}}$$ D. Zero

**Q8.** A circular wire loop of radius $$a$$ carries a total charge $$Q$$ distributed uniformly over its length. A small length $$d L$$ of the wire is cut off. Find the electric field at the centre due to the remaining wire.

A. Zero B. $$\cfrac{Q d L}{8 \pi^{2} \varepsilon_{0} a^{3}}$$

C. $$\cfrac{Q d L}{4 \pi^{2} \varepsilon_{0} a^{3}}$$ D. None of these

**Q9.** A thin wire ring has a charge $$q$$ uniformly spread on its circumference. A point charge $$Q$$ is placed at the centre of the ring. Find change in tension in the ring Radius of the ring is $$r$$.

A. $$k Q \sqrt{\cfrac{18}{5}}$$ B. $$k Q \sqrt{\cfrac{16}{5}}$$

C. $$2 k Q$$ D. $$\sqrt{2} k Q$$

**Q10.** A circular wire loop of radius $$R$$ carries a uniformly distributed charge $$Q$$ along its circumference. Two identical small segments of length $$\Delta L$$ are cut and removed from the wire. These segments are located at an angular separation of $$120^{\circ}$$ as shown in figure. Find electric field at the centre due to the remaining charge.

**Q11.** A particle of mass $$m$$ and charge $$-Q$$ is constrained to move along the axis of a ring of radius $$a$$. The ring has a uniform linear charge density $$\lambda$$ along its periphery. Initially, the particle is at the centre of the ring. It is displaced slightly and released.

Show that it will perform SHM and find time period of its oscillation.

**Q12.** A horizontal ring of radius $$a$$ has a uniformly spread negative charge $$Q$$ on it. A particle having mass $$m$$ is released at a height $$h$$ on the axis of the ring and it is found to stay at rest. Find the charge on the particle. $$h$$ is height measured from the plane of the ring.

**Q13.** A ring of charge with radius $$0.5\ m$$ has $$0.002\ \pi m$$ gap. If the ring carries a charge of $$+1\ C$$, the electric field at the center is

A. $$7.5 \times 10^{7} NC ^{-1}$$

B. $$7.2 \times 10^{7} NC ^{-1}$$

C. $$6.2 \times 10^{7} NC ^{-1}$$

D. $$6.5 \times 10^{7} NC ^{-1}$$

**Q14.** A thin glass rod is bent into a semicircle of radius $$r$$. A charge $$+Q$$ is uniformly distributed along the upper half, and a charge $$-Q$$ is uniformly distributed along the lower half, as shown in figure. The electric field $$E$$ at $$P$$, the center of the semicircle, is

A. $$\cfrac{Q}{\pi^{2} \varepsilon_{0} r^{2}}$$ B. $$\cfrac{2 Q}{\pi^{2} \varepsilon_{0} r^{2}}$$

C. $$\cfrac{4 Q}{\pi^{2} \varepsilon_{0} r^{2}}$$ D. $$\cfrac{Q}{4 \pi^{2} \varepsilon_{0} r^{2}}$$

**Q15.** A ring of radius $$R$$ has charge $$-Q$$ distributed uniformly over it. Calculate the charge $$(q)$$ that should be placed at the center of the ring such that the electric field becomes zero at a point on the axis of the ring distant $$R$$ from the center of the ring. Find the value of $$q$$.

**Q16.** A ring of radius $$0.1\ m$$ is made out of a thin metallic wire of area of cross section $$10^{-6} m ^{2}$$. The ring has a uniform charge of $$\pi$$ coulombs. Find the change in the radius of the ring when a charge of $$10^{-8} C$$ is placed at the center of the ring. Young's modulus of the metal is $$2 \times 10^{11} Nm ^{-2}$$.

**Q17.** A nonconducting ring of mass $$m$$ and radius $$R$$, with charge per unit length $$\lambda$$, is shown in figure. It is then placed on a rough nonconducting horizontal plane. At time $$t=0$$, a uniform electric field $$\vec{E}=E_{0} \hat{i}$$ is switched on and the ring starts rolling without sliding.

Determine the friction force (magnitude and direction) acting on the ring.

**Q18.** Two pieces of plastic, a full ring and a half ring, have the same radius and charge density. Which electric field at the center has the greater magnitude? Define your answer.

**Q19.** In the given figure $$A$$, a plastic rod in the form of circular arc with charge $$+Q$$ uniformly distributed on it produces an electric field of magnitude $$E$$ at the centre of curvature (at the origin).

In figures, $$B,\ C,\ D$$, more circular rods with identical uniform charges $$+Q$$ are added until the circle is complete. A fifth arrangement (which would be labeled $$E$$ ) is like that in $$D$$ except that the rod in the fourth quadrant has charge $$-Q$$.

Rank all the five arrangements according to the magnitude of the electric field at the center of curvature, greatest first.

**Q20.** A thin fixed ring of radius 1 meter has a positive charge $$1 \times 10^{-5} C$$ uniformly distributed over it. A particle of mass $$0.9\ g$$ and having a negative charge of $$1 \times 10^{-6}\ C$$ is placed on the axis at a distance of $$1\ cm$$ from the centre of the ring.

Show that the motion of the negatively charged particle is approximately simple harmonic. Calculate the time period of oscillations.

**Q21.** A wire of length $$L(=20 cm )$$, is bent into a semicircular arc. If the two equal halves of the arc were each to be uniformly charged with charges $$\pm Q,\left[|Q|=10^{3} \varepsilon_{0}\right.$$ Coulomb where $$\varepsilon_{0}$$ is the permittivity (in SI units) of free space] the net electric field at the centre $$O$$ of the semicircular arc would be

A. $$\left(50 \times 10^{3} N / C \right) \hat{j}$$

B. $$\left(50 \times 10^{3} N / C \right) \hat{i}$$

C. $$\left(25 \times 10^{3} N / C \right) \hat{j}$$

D. $$\left(25 \times 10^{3} N / C \right) \hat{i}$$

**Q22.** A thin non-conducting ring of radius $$a$$ has a linear charge density $$\lambda=\lambda_{0} \sin \theta$$. A uniform electric field $$E_{0} \hat{i}+E_{0} \hat{j}$$ is present there. Net torque acting on ring is

A. $$E_{0} \sqrt{2} \pi a^{2} \lambda_{0}$$ B. $$E_{0} \pi a^{2} \lambda_{0}$$

C. $$2 E_{0} \pi a^{2} \lambda_{0}$$ D. zero

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