Electric Forces & Fields

Multi-concept Questions

Multi-concept Questions

**Q1.** A wheel having mass $$m$$ has charges $$4 q$$ and $$-q$$ on diametrically opposite points. It remains in equilibrium on a rough inclined plane in the presence of uniform vertical electric field $$E$$. The value of $$E$$ is

A. $$\cfrac{m g}{q}$$ B. $$\cfrac{m g}{2 q}$$

C. $$\cfrac{m g \tan \theta}{2 q}$$ D. none of these

**Q2.** A particle of charge $$-q$$ and mass $$m$$ moves in a circle of radius $$r$$ around an in finitely long line charge of linear charge density $$+\lambda$$. Then time period will be given as

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

**Q3.** A copper (density of Cu $$=\rho_{C}$$ ) ball of diameter $$d$$ is immersed in oil of density $$\rho$$. What is the charge on the ball if, in a homogeneous electric field $$E$$ directed vertically upward, it is suspended in the oil ? $$\left(k \equiv \pi d^{3} \cfrac{\rho_{c} g}{E}\right)$$ :

A. $$\cfrac{1}{6} k\left(1-\cfrac{\rho_{0}}{\rho_{c}}\right)$$ B. $$\cfrac{1}{3} k\left(1-\cfrac{\rho_{0}}{\rho_{c}}\right)$$

C. $$\cfrac{1}{2} k\left(1-\cfrac{\rho_{0}}{\rho_{c}}\right)$$ D. $$k\left(1-\cfrac{\rho_{0}}{\rho_{c}}\right)$$

**Q4.** A small ball of mass $$m$$ and charge $$+q$$ tied with a string of length $$l$$, rotating in a vertical circle under gravity and a uniform horizontal electric field $$E$$ as shown in figure. The tension in the string will be minimum at an angle

A. $$\theta=\tan ^{-1}\left(\cfrac{q E}{m g}\right)$$ B. $$0=\pi$$

C. $$\theta=0^{\circ}$$ D. $$\theta=\pi+\tan ^{-1}\left(\cfrac{q E}{m g}\right)$$

**Q5.** Three semi-infinite rods uniformly charged out of which one is negatively charged and other two are positively charged one is negatively charged are kept perpendicular to plane of paper outward such that the finite ends of the rods are located at points $$A, B$$ and $$C$$ on a circle of radius $$R$$ as shown in figure. The net electric field at centre of circle $$O$$ is

A. $$\cfrac{2 K \lambda}{R}$$, along $$O C$$

B. $$\cfrac{K \lambda}{R}$$, perpendicular to plane of paper and inward direction

C. $$\cfrac{\sqrt{5} K \lambda}{R}$$ at an angle $$\tan ^{-1} \cfrac{1}{2}$$ with $$O C$$

D. $$\cfrac{\sqrt{2} K \lambda}{R}$$ at an angle $$45^{\circ}$$ with $$O C$$

**Q6.** A particle of mass $$m$$ and charge $$-q$$ is projected from the origin with a horizontal speed $$v$$ into an electric field of intensity $$E$$ directed downward. Choose the wrong statement. Neglect gravity.

A. The kinetic energy after a displacement $$y$$ is $$q E y$$.

B. The horizontal and vertical components of acceleration are $$a_{x}=q E / m, a_{y}=0$$.

C. The equation of trajectory is $$y^{2}=\cfrac{1}{2}\left(\cfrac{q E x^{2}}{m v^{2}}\right)$$.

D. The horizontal and vertical displacements $$x$$ and $$y$$ after time $$t$$ are $$x=v t^{2}$$ and $$y=\cfrac{1}{2} a_{y} t^{2}$$.

**Q7.** The position of two point charges $$q_{1}$$ and $$q_{2}$$ are $$\overrightarrow{r}_{1}$$ and $$\overrightarrow{r}_{2}$$. Find the position of point where net field is zero due to these charges?

a) $$\overrightarrow{r}=\cfrac{\overrightarrow{r}_{1} \sqrt{q_{2}}+\overrightarrow{r}_{2} \sqrt{q_{1}}}{\sqrt{q_{1}}+\sqrt{q_{2}}}$$

b.) $$\overrightarrow{r}=\cfrac{\overrightarrow{r}_{1} \sqrt{q_{1}}+\overrightarrow{r}_{2} \sqrt{q}_{2}}{\sqrt{q_{1}}+\sqrt{q}_{2}}$$

c) $$\overrightarrow{r}=\cfrac{\overrightarrow{r}_{1} q_{2}+\overrightarrow{r}_{2} a_{1}}{q_{1}+q_{2}}$$

d) $$\overrightarrow{r}=\cfrac{\overrightarrow{r}_{1} q_{1}+\overrightarrow{r}_{2} q_{2}}{q_{1}+q_{2}}$$

**Q8.** A tiny spherical oil drop carrying a net charge $$q$$ balanced in still air will a vertical uniform electric field of strength$$\cfrac{81 \pi}{7} \times 10^{5} V / m .$$ When the field is switched off, the drop is observed to fall with terminal velocity $$2 \times 10^{-3} m / s$$.

Given $$g=9.8 m / s ^{2}$$, viscosity of the air $$=1.8 \times 10^{-5} Ns / m ^{2}$$ and the density of oil $$=900 kg / m ^{3}$$, the magnitude of $$q$$ is

A. $$1.6 \times 10^{-19} C$$ B. $$3.2 \times 10^{-19} C$$

C. $$4.8 \times 10^{-19} C$$ D. $$7.8 \times 10^{-19} C$$

**Q9.** The two ends of a rubber string of negligible mass and having unstretched length $$24\ cm$$ are fixed at the same heightar shown. A small object is attached to the string in its midpoint due to which the depression $$h$$ of the object in equilibrium is $$5\ cm$$. Then the small object is charged and a vertical electric field $$E_{1}$$ is switched on in the region.

The equilibrium depression of the object increases to $$9\ cm$$, now the electric field is changed to $$E_{2}$$ and the depression of object in equilibrium increases to $$16\ cm$$. What is the ratio of electric field in the second caset that of in the first case?

A. $$4.25$$ B. $$4.20$$ C. $$4.30$$ D. $$4.35$$

**Q10.** Let there be a spherically symmetric charge distribution with charge density varies with distance $$r$$ from the center and given as

$$\rho(r)=\rho_{0}\left(\cfrac{5}{4}-\cfrac{r}{R}\right)$$ for $$r=R $$

and $$\rho(r)=0$$ for $$r>R$$.

The electric field at a distance $$r(r

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

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

**Q11.** An electric dipole is placed perpendicular to an infinite line of charge at some distance as shown in figure. Identify the correct statement(s).

A. The dipole is attracted towards the line charge.

B. The dipole is repelled away from the line charge.

C. The dipole does not experience a force.

D. The dipole experiences a force as well as a torque.

**Q12.** A particle of mass $$2\ kg$$ and charge $$1\ mC$$ is projected vertically with a velocity $$10\ m / s$$. There is a uniform horizontal electric field of $$10^{4} N / C$$.

A. The horizontal range of the particle is $$10\ m$$.

B. The time of flight of the particle is $$2\ s$$.

C. The maximum height reached is $$5\ m$$.

D. The horizontal range of the particle is $$5\ m$$.

**Q13.** The following figure shows a block of mass $$m$$ suspended from a fixed point by means of a vertical spring. The block is oscillating simple harmonically and carries a charge $$q$$. There also exists a uniform electric field in the space. Consider four different cases.

The electric field is zero, in case - 1,

$$E=m g / q$$ downward in case - 2,

$$E=m g / q$$ upward in case - 3,

and $$E=2 m g / q$$ downward in case - 4.

The speed at mean position of block is same in all cases. Select which of the following statements is/are correct.

A. Time periods of oscillation are equal in case - 1 and case - 3.

B. Amplitudes of displacement are same in case - 2 and case - 3.

C. The maximum elongation (increment in length from natural length) is maximum in case - 4.

D. Time periods of oscillation are equal in case - 2 and case - 4.

**Q14.** An insulating rod of uniform linear charge density $$\lambda$$ and uniform linear mass density $$\mu$$ lies on a smooth table whose surface is $$x y$$ -plane. A uniform electric field $$E$$ is switched on in the space.

A. If electric field is along $$x$$ -axis, the speed of the rod when it has travelled a distance $$d$$ is $$\sqrt{\cfrac{2 \lambda E d}{\mu}}$$.

B. If electric field $$E$$ is at an angle $$\theta\left(<90^{\circ}\right)$$ with $$x$$ -axis along the table surface then the speed of the rod when it has travelled a distance $$d$$ is $$\sqrt{\cfrac{2 \lambda E d \cos \theta}{\mu}}$$.

C. A non zero torque acts on the rod due to the field about centre of mass in case electric field is into the plane of paper.

D. A non zero torque acts on the rod due to the field about centre of mass in case electric field is along the surface of table.

**Q15.** A charge $$q$$ is revolving around another charge $$q$$ as shown in a conical pendulum. The motion is in a horizontal plane. Which of the following statements is/are correct about this situation.

A. Tension in the string is greater than the weight of the ball.

B. The tension in the string is greater than the electrostatic repulsive force.

C. If the charge is removed, the speed of the ball has to be increased to maintain the angle.

D. If the charge is removed, the speed of ball has to be decreased to maintain the angle.

**Q16.** An electric dipole is placed in an electric filed generated by a point charge.

A. The net force on the dipole never be zero.

B. The net force on the dipole may be zero.

C. The torque on the dipole due to the field must be zero.

D. The torque on the dipole due to the field may be zero.

**Q17.** Two large thin conducting plates with small gap in between are placed in an uniform electric field $$E$$ which exist in the direction as shown in figure-1.433. Area of each plate is $$A$$ and charges $$+Q$$ and $$-Q$$ are given to those plates as shown in the figure. If points $$R, S$$ and $$T$$ are three points in space, then which of the following is/are correct ?

A. Field at point $$R$$ is $$E$$.

B. Field at point $$S$$ is $$E$$.

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

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

**Q18.** Three non-conducting infinite planar sheets are parallel to the $$y-z$$ plane. Each sheet has an uniform surface charge density density. The first sheet, with a negative surface charge $$-\sigma$$, passes through the $$x$$ -axis at $$x=1\ m$$. The second sheet has through an unknown surface charge density and passes axis at $$x=2\ m$$. The third sheet has a negative surface charge density $$-3 \sigma$$ and passes through the $$x$$ -axis at $$x=4\ m$$. The net electric field due to the sheets is zero at $$x=1.5\ m$$.

Which of the following is/are correct:

A. The surface charge density on the second sheet is $$+2 \sigma$$.

B. The electric field at $$x=-2 m$$ is $$\cfrac{\sigma}{\epsilon_{0}} \hat{i}$$.

C. The electric field at $$x=3 m$$ is $$\cfrac{\sigma}{\epsilon_{0}} \hat{i}$$.

D. The electric field at $$x=6 m$$ is $$\cfrac{-\sigma}{\epsilon_{0}} \hat{i}$$.

**Q19.** A particle of mass $$1.6 \times 10^{-30} {~kg}$$ and charge $$1.6 \times 10^{-19} {C}$$ is projected with initial an initial speed $$u$$ at an angle of $$45^{\circ}$$ to the horizontal from lower plate of a parallel plate capacitor as shown. Find the maximum velocity of particle so that it does not hit the upper plate. Take $$E=10^{3} {~V} / {m}$$ directed downward.

**Q20.** A wooden block performs SHM on a frictionless surface with frequency $$v_{0}$$. The block carries a charge $$+Q$$ on its surface. If now a uniform electric field $$\vec{E}$$ is switched-on as shown, then the SHM of the block will be

A. of the same frequency and with shifted mean position.

B. of the same frequency and with the same mean position.

C. of changed frequency and with shifted mean position.

D. of changed frequency and with the same mean position.

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