# Numerical Problems : Electrostatics

#### LEVEL – I

Q:1. Point charges of magnitude q, 2q and 8q are to be placed on a 9 cm long straight line. Find the positions where the charges should be placed such that potential energy of this system is minimum.

[Ans: 2q , 8q at the two ends and q at 3cm from 2q ]

Q:2. Water from a metal vessel maintained at a potential of 3 volt falls in spherical drops 2 mm in diameter through a small hole into a thin walled isolated metal sphere of diameter 8 cm placed in air until the sphere is filled with water. Ignoring the thickness of the metal calculate the final potential of the sphere and its electrical energy.

[Ans: 4800 V, 512 × 10-7 J]

Q:3. An infinite number of charges each equal to ‘ q ‘ are placed along the x-axis at x = 1, x = 2, x = 4, x = 8, and so on. Find the potential and electric field at the point x = 0 due to this set of charges. What will be the potential and electric field if in the above set up the consecutive charges have opposite sign?

Q:Ans: $\displaystyle \frac{4 k q}{3}$

Q:4. A uniform electric field of strength 106 V/m is directed vertically downwards. A particle of mass 0.01 kg and charge 10-6 coulomb is suspended by an inextensible thread of length 1m. The particle is displaced slightly from its mean position and released.

(a) Calculate the time period of its oscillation.

(b) What minimum velocity should be given to the particle at rest so that it completes a full circle in a vertical plane without the thread getting slack?

(c) Calculate the maximum and minimum tensions in the thread in this situation.

[Ans: (a) 0.6 sec (b) 23.42 m/s (c) 6.588, Zero ]

Q:5. Two equal charges q are kept fixed at −a and +a along the x-axis . A particle of mass m and charge (q/2)is brought to the origin and given a small displacement along the (a) X-axis and (b) Y-axis. Describe quantitatively the motion in two cases.

[Ans: (a) SHM , (b) continue to move up along the Y-axis]

Q:6. A strip of length ‘ l ‘ having linear charge density ‘σ ‘ is placed near a negatively charged particle ‘ P ‘ of mass ‘ m ‘ and charge ‘ -q ‘ (as shown in the figure) at a distance ‘ d ‘ from the end ‘ A ‘ of the strip. Find the velocity of ‘ P ‘ as it reaches a point at the distance d/2 from end ‘ A ‘ . Ans: $\displaystyle v = \sqrt{\frac{\sigma q}{2\pi \epsilon_o m} (\frac{d+2l}{d+l})}$

Q:7. A thin fixed ring of radius ‘ R ‘ and positive charge ‘ Q ‘ is placed in a vertical plane. A particle of mass ‘ m ‘ and charge ‘ q ‘ is placed at the centre of ring. If the particle is given a small horizontal displacement, show that it executes SHM also find the time period of small oscillations of this particle, about the centre of ring. (Ignore gravity)

Ans: $\displaystyle T = 2\pi \sqrt{\frac{4 \pi \epsilon_0 m R^3}{q Q}}$

Q:8. A non-conducting sphere having a cavity as shown in figure is uniformly charged with volume charge density ρ. Find the potential at a point P which is at a distance of x from C. Ans: $\displaystyle \frac{4}{3}\pi R^3 k \rho (\frac{1}{x} – \frac{1}{\sqrt{x^2 +R^2/4}})$

Q:9. A particle of charge q and mass m moves along the x-axis under the action of an electric field
E = k − c x , where ‘ c ‘ is a positive constant and x is distance from the point, where particle was initially at rest.

Calculate :
(a) distance travelled by the particle before it comes to rest.

(b) acceleration at the moment , when it comes to rest.

Ans: (a)2k/c (b) – q k/m

Q:10. Charges +q and −q are located at the corners of a cuboid as shown in the figure. Find the electric potential energy of the system. Ans: $\displaystyle P.E = \frac{k q^2}{a}[2\sqrt{2} – 4 (1+ \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{5}})]$

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