Electric Forces & Fields
Electric Field and Neutral Points
DPP # EFF-03

Q1. A wooden ball covered with an aluminum foil having a mass $$m$$ hangs by a fine silk thread $$l$$ meter long in a horizontal electric field $$E$$. When the ball is given an electric charge $$q$$ coulomb, it stands out $$d$$ meter from the vertical line passing through the suspension point of thread.

Show that the electric field is given by

$$E=\cfrac{m g d}{q \sqrt{l^{2}-d^{2}}}$$.

Q2. A clock face has charges $$-q,-2 q,-3 q, \ldots-12 q$$ fixed at the position of the corresponding numerals on the dial. The clock hands do not disturb the net field due to point charges.

At what time does the hour hand point in the direction of the electric field at the center of the dial?

Q3. The intensity of the electric field required to keep a water drop of radius $$10^{-5} cm$$ just suspended in air when charged with one electron is approximately $$(g=10 N / kg$$, $$\left.e=1.6 \times 10^{-19} C \right)$$

A. $$260 \ V / cm$$     B. $$260\ N / C$$

C. $$130\ V / cm$$    D. $$130\ N / C$$

Q4. Four charges are placed on corners of a square as shown in the figure having side of $$5\ cm$$. If $$Q$$ is one microcoulomb, then electric field intensity at the center of the square will be

A. $$1.02 \times 10^{7} N / C , \ upwards$$

B. $$2.04 \times 10^{7} N / C, \ downwards$$

C. $$2.04 \times 10^{7} N / C, \ upwards$$

D. $$1.02 \times 10^{7} N / C, \ downwards$$

Q5. Three identical point charges, as shown in the figure, are placed at the vertices of an isosceles right angled triangle. Which of the numbered vectors coincides in direction with the electric field at the mid-point $$M$$ of the hypotenuse?

A. 1     B. 2     C. 3     D. 4

Q6. A simple pendulum is suspended in a lift which is going up with an acceleration of $$5 \ m / s ^{2}$$. An electric field of magnitude $$5 \ N / C$$ and directed vertically upward is also present in the lift. The charge of the bob is $$1 \ \mu C$$ and mass is $$1 \ mg$$.

Taking $$g=\pi^{2}$$ and length of the simple pendulum $$1 \ m$$, find the time period of the simple pendulum (in seconds).

A. $$2\ s$$     B. $$4\ s$$     C. $$1\ s$$     D. $$5\ s$$

Q7. A point charge $$50 \mu C$$ is located in an $$X Y$$ plane at the point of position vector $$\vec{r}_{0}=(2 \hat{i}+3 \hat{j}) m$$. What is the electric field at the point of position vector $$\vec{r}=(8 \hat{i}-5 \hat{j}) m ?$$

A. $$1200\ V / m$$     B. $$0.04\ V / m$$

C. $$900\ V / m$$    D. $$4500\ V / m$$

Q8. An electron of mass $$m_{e}$$, initially at rest, moves through a certain distance in a uniform electric field in time $$t_{1}$$. A proton of mass $$m_{p}$$, also, initially at rest, takes time $$t_{2}$$ to move through an equal distance in this uniform electric field. Neglecting the effect of gravity, the ratio $$\cfrac{t_{2}}{t_{1}}$$ is nearly equal to

A. 1   B. $$\left(\cfrac{m_{p}}{m_{e}}\right)^{{1}/{2}}$$ C. $$\left(\cfrac{m_{e}}{m_{p}}\right)^{{1}/{2}}$$   D. 1836

Q9. If a charge $$16Q$$ is at point $$A$$ and a charge $$25 Q$$ is at point $$B$$, then calculate $$|\vec{E}|$$ at origin.

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

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

Q10. Find the magnitude of uniform electric field $$E$$ (in $$N / C$$ ) (direction shown in figure) if an electron entering with velocity $$100\ m / s$$ and making $$30^{\circ}$$ with $$x$$ -axis comes out making $$60^{\circ}$$ with $$x$$ -axis after a time numerically equal to $$\cfrac{m}{e}$$ of electron.

A. 100     B. 150     C. 200     D. 250

Q11. A simple pendulum having bob of mass $$m$$ and charge $$+q$$ has length $$\ell$$. It is placed in downward vertical electric field. The time period of small oscillation is

A. equal to $$2 \pi \sqrt{\cfrac{\ell}{g}}$$     B. greater than $$2 \pi \sqrt{\cfrac{\ell}{g}}$$

C. less than $$2 \pi \sqrt{\cfrac{\ell}{g}}$$     D. not defined

Q12. Infinite charges of magnitude $$q$$ each are lying at $$x=1,2$$, 4, $$8 \ldots$$ meter on $$X$$ -axis. The value of intensity of electric field at point $$x=0$$ due to these charges will be

A. $$12 \times 10^{9} q N / C$$     B. Zero

C. $$6 \times 10^{9} q N / C$$     D. $$4 \times 10^{9}$$ q $$N / C$$

Q13. Two identical point charges are placed at a separation of $$d$$. $$P$$ is a point on the line joining the charges, at a distance $$x$$ from any one charge. The field at $$P$$ is $$E,\ E$$ is plotted against $$x$$ for values of $$x$$ from close to zero to slightly less than $$d$$. Which of the following represents the resulting curve?

Q14. Two charges each equal to $$\eta q\left(\eta^{-1}<\sqrt{3}\right)$$ are placed at the corners of an equilateral triangle of side $$a$$. The electric field at the third corner is $$E_{3}$$ where $$\left(E_{0}=q / 4 \pi \varepsilon_{0} a^{2}\right)$$

A. $$E_{3}=E_{0}$$     B. $$E_{3} \lt E_{0}$$

C. $$E_{3} \gt E_{0}$$    D. $$E_{3} \geq E_{0}$$

Q15. The following figures show regular hexagons, with charges at the vertices. In which of the following cases the electric field at the centre is not zero?

A. 1     B. 2     C. 3    D. 4

Q16. When two point charges $$q_{A}$$ and $$q_{B}$$ are place at some separation on positive $$x$$ -axis at points $$\left(x_{A}, 0\right)$$ and $$\left(x_{B}, 0\right)$$. Given that $$\left|q_{A}\right|\gt\left|q_{B}\right|$$ and $$x_{B}\gtx_{A}$$. If null point is the point where net electric field due to both the charges is zero, then

A. if both $$q_{A}$$ and $$q_{B}$$ are positive, null point lies at some point $$x_{A} B. if$$q_{A}$$is positive and$$q_{B}$$is negative, null point lies at some point$$x

C. if $$q_{A}$$ is positive and $$q_{B}$$ is negative, null point lies at some point $$x\gtx_{B}$$

D. if $$q_{A}$$ is negative and $$q_{B}$$ is positive, null point lies at some point $$x\gtx_{B}$$

Q17. In the following four situations charged particles are at equal distance from the origin. Arrange them according to the magnitude of the net electric field at origin greatest first.

A. $$(i) \gt (ii) \gt (iii) \gt (iv)$$

B. $$(ii) \gt( i )\gt( iii )\gt( iv )$$

C. $$( i )\gt( iii )\gt( ii )\gt( iv )$$

D. $$(iv) \gt( iii )\gt( ii )\gt( i )$$

Q18. A uniform electric field $$E$$ exists between two metal plates one negative and other positive. The plate length is $$l$$ and the separation of the plates is $$d$$.

(i) An electron and a proton start from the negative plate and positive plate, respectively, and go to opposite plates. Which one of them wins this race?

(ii) An electron and a proton start moving parallel to the plates toward the other end from the midpoint of the separation of plates at one end of the plates. Which of the two will have greater deviation when they come out of the plates if they start with the

a. same initial velocity,

b. same initial kinetic energy, and

c. same initial momentum.

Q19. An electron (mass $$m_{e}$$ ) falls through a distance $$d$$ in a uniform electric field of magnitude $$E$$.

The direction of the field is reversed keeping its magnitude unchanged, and a proton (mass $$m_{p}$$ ) falls through the same distance. If the times taken by the electron and the proton to fall the distance $$d$$ is $$t_{\text {electron }}$$ and $$t_{\text {proton }}$$, respectively, then the ratio $$t_{\text {electron }} / t_{\text {proton }}$$ is equal to

Q20. Three charged particles $$A , B$$ and $$C$$ with charges $$-4 q,\ 2 q$$ and $$-2 q$$ are present on the circumference of a circle of radius $$d$$. The charged particles $$A , C$$ and centre $$O$$ of the circle formed an equilateral triangle as shown in figure. Electric field at $$O$$ along $$x$$ -direction is

A. $$\cfrac{\sqrt{3} q}{4 \pi \varepsilon_{0} d^{2}}$$     B. $$\cfrac{3 \sqrt{3} q}{4 \pi \varepsilon_{0} d^{2}}$$

C. $$\cfrac{\sqrt{3} q}{\pi \varepsilon_{0} d^{2}}$$     D. $$\cfrac{2 \sqrt{3} q}{\pi \varepsilon_{0} d^{2}}$$

Q21. A simple pendulum of length $$L$$ is placed between the plates of a parallel plate capacitor having electric field $$E$$, as shown in figure. Its bob has mass $$m$$ and charge $$q .$$ The time period of the pendulum is given by:

A. $$2 \pi \sqrt{\cfrac{L}{\left(g+\cfrac{q E}{m}\right)}}$$

B. $$2 \pi \sqrt{\cfrac{L}{ \left.\sqrt{\left(g^{2}-\cfrac{q^{2} E^{2}}{m^{2}}\right.}\right)}}$$

C. $$2 \pi \sqrt{\cfrac{L}{\left(g-\cfrac{q E}{m}\right)}}$$

D. $$2 \pi \sqrt{\cfrac{ L }{\sqrt{g^{2}+\left(\cfrac{q E }{m}\right)^{2}}}}$$

Q22. A particle of mass $$m$$ and charge $$q$$ is released from rest in a uniform electric field. If there is no other force on the particle, the dependence of its speed $$v$$ on the distance $$x$$ travelled by it is correctly given by (graphs are schematic and not drawn to scale)

Q23. A small point mass carrying some positive charge on it, is released from the edge of a table. There is a uniform electric field in this region in the horizontal direction. Which of the following options then correctly describe the trajectory of the mass? (Curves are drawn schematically and are not to scale)

Q24. A charged particle (mass $$m$$ and charge $$q$$ ) moves along $$X$$ axis with velocity $$V_{0} .$$ When it passes through the origin it enters a region having uniform electric field $$\overrightarrow{ E }=- E \hat{j}$$ which extends upto $$x=d$$. Equation of path of electron in the region $$x \gt d$$ is

A. $$y=\cfrac{q E d}{m V _{0}^{2}} x$$

B. $$y=\cfrac{q E d}{m V _{0}^{2}}(x-d)$$

C. $$y=\cfrac{q E d}{m V _{0}^{2}}\left(\frac{d}{2}-x\right)$$

D. $$y=\cfrac{q E d^{2}}{m V_{0}^{2}} x$$

Q25. Two point charges $$q_{1}(\sqrt{10} \mu C)$$ and $$q_{2}(-25 \mu C)$$ are placed on the $$x$$ -axis at $$x=1 m$$ and $$x=4 m$$ respectively. The electric field (in $$V / m$$ ) at a point $$y=3 m$$ on $$y$$ -axis is, $$\left(\right.$$ take $$\left.\cfrac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} Nm ^{2} C ^{-2}\right)$$

A. $$(63 \hat{i}-27 \hat{j}) \times 10^{2}$$

B. $$(-63 \hat{i}+27 \hat{j}) \times 10^{2}$$

C. $$(81 \hat{i}-81 \hat{j}) \times 10^{2}$$

D. $$(-81 \hat{i}+81 \hat{j}) \times 10^{2}$$