07-Numerical Problems

“Electrostatic” Numerical Problems

Try to solve the following “Electrostatic” numerical problems to clear the concepts in solving the numerical problems.

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01. PROBLEM – P070101

Two identical spheres $A$ and $B$ each having charge $( 6.5 \times 10^{-7} C )$ are separated by a distance of $( 50 \ cm )$ in air. A third uncharged sphere of same size is brought in contact with the first, then brought in contact with the second and afterwards removed from both. What is the new electrical force between the first two spheres?

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02. PROBLEM – P070102

An electron falls through a distance of $( 4 \ cm )$ in a uniform electric field of value $( 5 \times 10^{4} )$ . When the direction of the field is reversed, a proton falls through the same distance. Calculate the (i) acceleration, (ii) time of fall in each case.

Explain why the effect of gravity is not considered in such case.

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03. PROBLEM – P070103

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of $30 \degree$ with each other. When this set up is suspended in a liquid of density $( 0.8 \ gm \ cm^{-3} )$ , the angle remains the same.

DIELECTRIC CONSTANT OF LIQUID — PROBLEM FIGURE P070103

What is the dielectric constant of the liquid? Given – density of the material of the sphere is $( 1.6 \ gm \ cm^{-3} )$ .

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04. PROBLEM – P070104

Two small balls, each of mass m and carrying equal charges are suspended from a point through silk threads of length $( l )$ each. The distance between the balls in the steady state is $( x )$ where $( x << l )$ .

COULOMB'S CONSTANT FOR CHARGED PARTICLE — PROBLEM FIGURE P070104

At what rate the charges from the balls should leak so that the balls may attract each other with a relative velocity, $\left [ v = \left ( \frac {a}{\sqrt {x}} \right ) \right ]$ where $( a )$ is a constant. Take coulomb’s constant as $( k )$ .

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05. PROBLEM – P070105

A charge of $( 4 \times 10^{- 8} C )$ is distributed uniformly on the surface of a sphere of radius $( 1 \ cm )$ . It is covered by a concentric, hollow conducting sphere of radius $( 5 \ cm )$ .

Find the electric field at a point $( 2 \ cm )$ away from the centre.
A charge of $( 6 \times 10^{- 8} C )$ is placed on the hollow sphere. Find the surface charge density on the outer surface of the hollow sphere.

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06. PROBLEM – P070106

Three concentric thin spherical shells $A \ B \ \& \ C$ of radii $( a ), \ ( b ) \ \& \ ( c )$ respectively are placed as shown in figure. The shells $A$ and $C$ are given charges $( q )$ and $( - q )$ respectively and the shell $B$ is earthed.

DISTRIBUTION OF CHARGE ON THREE CONCENTRIC THIN SPHERICAL SHELLS — PROBLEM FIGURE P070106

Find the charges appearing on the surfaces of $B$ and $C$ .

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07. PROBLEM – P070107

Two conducting plates $A$ and $B$ are placed parallel to each other. A is given a charge $( Q_1 )$ and $B$ a charge $( Q_2 )$ .

DISTRIBUTION OF CHARGE ON PARALLEL PLATES — PROBLEM FIGURE P070107

Find the distribution of charges on the four surfaces.

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08. PROBLEM – P070108

A charge $( Q )$ is placed at a distance of $\left ( \frac {a}{2} \right )$ above the centre of a horizontal square surface of edge $( a )$ as shown in figure.

GAUSS LAW FOR FLUX THROUGH THE RECTANGULAR PLATE — PROBLEM FIGURE P070108

Find the flux of the electric field through the square surface.

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09. PROBLEM – P070109

A charge $( Q )$ is placed at the centre of an imaginary hemispherical surface.

FLUX DISTRIBUTION FOR CHARGE PLACED AT CENTER OF A HEMISPHERE — PROBLEM FIGURE P070109

Using symmetry arguments and the Gauss law, find the flux of the electric field due to this charge through the surface of the hemisphere.

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10. PROBLEM – P070110

A charge of $( 2 \times 10^{- 9} C )$ is placed on a corner of a cube of side $( 1 \ m )$ as shown in figure.

DISTRIBUTION OF FLUX FOR A CHARGE PLACED AT THE CORNER OF A CUBE — PROBLEM FIGURE P070110

Find the (a) Electric flux passing through this cube (b) Find the flux passing through a face of the given cube.

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11. PROBLEM – P070111

Two identical parallel plate capacitors $A$ and $B$ are connected to a battery of $( V )$ volt with the switch $S$ closed as shown in figure. The switch is now opened and the free space between the plates of the capacitors is filled with a dielectric medium of dielectric constant $( K )$ .

ENERGY STORED IN PARALLEL PLATE CAPACITOR — PROBLEM FIGURE P070111

Find the ratio of the total electrostatic energy stored in both capacitors before and after the introduction of the dielectric.

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12. PROBLEM – P070112

Electric field at a point due to a point charge is $( 20 \ N \ C^{- 1} )$ and the electric potential at that point is, $( 10 \ J \ C^{- 1} )$ . Calculate the distance of the point from the charge and the magnitude of the charge.

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13. PROBLEM – P070113

Three charges $( - q ), \ ( + Q ) \ \& \ ( - q )$ are placed at equal distances on a straight line. If the total potential energy of the system is zero, then find the ratio $\left ( \frac {Q}{q} \right )$

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14. PROBLEM – P070114

A student requires a capacitor of $( 3 \mu F )$ in a circuit across a potential of $( 1000 \ V )$ . A large number of $( 2 \mu F )$ capacitors are available to him each of which can withstand a potential difference of not more than $( 300 \ V )$ .

How the student should arrange these capacitors so that he may need minimum number of capacitors?

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15. PROBLEM – P070115

Two circular loops of radius $( 0.5 \ m )$ and $( 0.09 \ m )$ respectively are put such that their axis coincides and their centres are $( 0.12 \ m )$ apart as shown in figure. Charge of $( 1 \times 10^{- 6} C )$ is spread uniformly on each loop.

POTENTIAL DIFFERENCE BETWEEN TWO CONCENTRIC CIRCULAR LOOPS — PROBLEM FIGURE P070115

Find the potential difference between the centres of loops.

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16. PROBLEM – P070116

Find the equivalent capacitance between the terminals $A$ and $B$ in the given figure. Given, $( C = 1 \mu F )$

EQUIVALENT CAPACITANCE OF MIXED COMBINATION OF CAPACITORS — PROBLEM FIGURE P070116

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17. PROBLEM – P070117

Calculate the capacitance between the points $A$ and $B$ of the arrangement shown in the figure (A) and (B), if the area of each plate is $( A )$ and distance between successive plates is $( d )$ .

EQUIVALENT CAPACITANCE — PROBLEM FIGURE P070117

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18. PROBLEM – P070118

How many electrons per second flow through the cross section of a conductor so that the conductor carries a current of $( 1 \ A )$ ampere?

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Electric current.

19. PROBLEM – P070119

In an atom, an electron revolves round the nucleus in a circular orbit at the rate of $( 10^5 )$ revolutions per second. Calculate the equivalent current. Take $( e = 1.6 \times 10^{-19} \ C )$

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Electric current.

20. PROBLEM – P070120

(a) Estimate the average drift speed of conduction electrons in a copper wire of cross sectional area $( 1.0 \times 10^{-7} \ m^2 )$ carrying a current of $( 1.5 \ A )$ . Assume that each copper atom contributes one conduction electron. The density of copper is $( 9.0 \times 10^3 \ kg-m^3 )$ and its atomic mass is $( 63.5 \ u )$

(b) Compare the drift speed obtained above with (i) thermal speed of electrons carrying the current and (ii) speed of propagation of electric field along the conductor which causes the drift motion.

Given, Avogadro’s number is $( N = 6.0 \times 10^{23} )$ per kg – atom. Boltzmann constant is $( k = 1.38 \times 10^{-23} \ J K^{-1} )$ , mass of electron is $( 9.1 \times 10^{-31} \ kg )$

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21. PROBLEM – P070121

In a discharge tube, the number of hydrogen ions (i.e. protons) drifting across a cross section per second is $( 1.0 \times 10^{18} )$ , while the number of electrons drifting in the opposite direction across that cross section is $( 2.7 \times 10^{18} )$ per second. If the supply voltage is $( 230 \ V )$ , what is the effective resistance of the tube?

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22. PROBLEM – P070122

A resistor of $( 24 \ \Omega )$ resistance is bent in the form of a circle as shown in figure. What is the effective resistance between points $A$ and $B$ ?

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23. PROBLEM – P070123

A $( 500 \ W )$ lamp gives light for $10$ hours on electric mains. Find the cost of electric consumption for $30$ days if cost of energy is $( Rs. \ 3.00 )$ per unit.

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24. PROBLEM – P070124

A given wire is stretched to reduce its diameter to half of its original value. What will be its new resistance?

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25. PROBLEM – P070125

Find the internal resistance of a cell having e.m.f of $( 2 \ V )$ . The potential difference across the terminals of this cell drops to $( 1 \ V )$ when a resistance of $( 10 \ \Omega )$ is connected across it.

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26. PROBLEM – P070126

Calculate the potential difference between $B$ and $D$ points in the given figure. An e.m.f of $( 12 \ V )$ is connected in the circuit.

EQUIVALENT RESISTANCE OF MIXED COMBINATION OF RESISTANCE — PROBLEM FIGURE P070126

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27. PROBLEM – P070127

A battery of e.m.f $( 10 \ V )$ and internal resistance $( 0.5 \ \Omega )$ is charged by a d.c. source of $( 12 \ V )$ with the help of a series resistor of $( 9.5 \ \Omega )$ . Find the terminal voltage of the battery when it is being charged.

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28. PROBLEM – P070128

Two cells of e.m.f of $( 1 \ V )$ and $( 2 \ V )$ and internal resistances of $( 2 \ \Omega )$ and $( 1 \ \Omega )$ respectively, are connected in (i) series (ii) parallel. What should be the external resistance $( R )$ in the circuit so that the current through the resistance be the same in the two cases?

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29. PROBLEM – P070129

Find the equivalent resistance between the points $a \ \& \ c$ of the network shown in figure. Each resistance is equal to $( r )$ .

EQUIVALENT RESISTANCE OF MIXED COMBINATION OF RESISTORS — PROBLEM FIGURE P070129

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30. PROBLEM – P070130

Twelve wires, each of resistance $( r )$ ohms are connected in the form of a skeleton cube. Find the equivalent resistance of the cube when the current enters at one corner and leaves at the diagonally opposite corner.

EQUIVALENT RESISTANCE OF A CUBE OF RESISTOR WIRES — PROBLEM FIGURE P070130

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31. PROBLEM – P070131

Twelve wires, each of resistance $( r )$ ohms are connected in the form of a skeleton cube. Find the equivalent resistance between the ends of a face diagonal such as $A \ \& \ C$ .

EQUIVALENT RESISTANCE OF 12 RESISTORS IN A CUBE — PROBLEM FIGURE P070131

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32. PROBLEM – P070132

Twelve wires, each of resistance $( r )$ ohms are connected in the form of a skeleton cube. Find the equivalent resistance between the ends of an edge such as $A \ \& \ B$ .

TWELVE RESISTANCES IN FORM OF A SKELETON OF CUBE — PROBLEM FIGURE P070132

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33. PROBLEM – P070133

Find the currents in the different resistors shown in figure.

CURRENT IN A CIRCUIT — PROBLEM FIGURE P070133

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34. PROBLEM – P070134

In the circuit shown in figure $E, \ F, \ G \ \& \ H$ are cells of e.m.f $( 2 \ V ), \ ( 1 \ V ), \ ( 3 \ V ) \ \& \ ( 1 \ V )$ respectively. The resistances $( 2 \ \Omega ), \ ( 1 \ \Omega ), \ ( 3 \ \Omega ) \ \& \ ( 1 \ \Omega )$ are their respective internal resistances. Calculate (a) the potential difference between $B$ and $D$ and (b) the potential differences across the terminals of each of the cells $G$ and $H$ .

CURRENT IN A RESISTANCE CIRCUIT — PROBLEM FIGURE P070134

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35. PROBLEM – P070135

A part of a circuit in steady state along with the currents flowing in the branches, the values of resistances etc. is shown in figure.

EQUIVALENT EMF IN A CIRCUIT — PROBLEM FIGURE P070135

Calculate the energy stored in the Capacitor.

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36. PROBLEM – P070136

In the meter bridge experimental set up shown in figure, the null point $'D'$ is obtained at a distance of $( 40 \ cm )$ from end $A$ of the meter bridge wire. If a resistance of $( 10 \ \Omega )$ is connected in series with $( R_1 )$ , null point is obtained at $AD = ( 60 \ cm )$ . Calculate the values of $( R_1 ) \ \& \ ( R_2 )$ .

FINDING UNKNOWN RESISTANCE WITH METER BRIDGE — PROBLEM FIGURE P070136

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37. PROBLEM – P070137

In the given circuit $AB$ is a uniform wire of $( 10 \ \Omega )$ and length $( 1 \ m )$ . It is connected to series arrangement of cell $E_1$ of e.m.f $( 2 \ V )$ and negligible internal resistance and a resistor $( R )$ . Terminal $A$ is also connected to an electro-chemical cell $E_2$ of e.m.f $( 100 \ mV )$ and a galvanometer $G$ . In this set up a balancing point is obtained at $( 40 \ cm )$ mark from $A$ . Calculate the resistance $( R )$ . If $E_2$ were to have an e.m.f of $( 300 \ mV )$ , where will you expect the balancing point to be?

NULL POINT OF A METER BRIDGE — PROBLEM FIGURE P070137

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38. PROBLEM – P070138

Find the equivalent resistance between the points $( a ) \ \& \ ( b )$ of the infinite ladder shown in figure.

EQUIVALENT RESISTANCE OF A LADDER OF INFINITE IDENTICAL RESISTANCES — PROBLEM FIGURE P070138

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Numericals » 07-Numerical Problems

“Electrostatic” Numerical Problems

01. PROBLEM – P070101

02. PROBLEM – P070102

03. PROBLEM – P070103

04. PROBLEM – P070104

05. PROBLEM – P070105

06. PROBLEM – P070106

07. PROBLEM – P070107

08. PROBLEM – P070108

09. PROBLEM – P070109

10. PROBLEM – P070110

11. PROBLEM – P070111

12. PROBLEM – P070112

13. PROBLEM – P070113

14. PROBLEM – P070114

15. PROBLEM – P070115

16. PROBLEM – P070116

17. PROBLEM – P070117

18. PROBLEM – P070118

19. PROBLEM – P070119

20. PROBLEM – P070120

21. PROBLEM – P070121

22. PROBLEM – P070122

23. PROBLEM – P070123

24. PROBLEM – P070124

25. PROBLEM – P070125

26. PROBLEM – P070126

27. PROBLEM – P070127

28. PROBLEM – P070128

29. PROBLEM – P070129

30. PROBLEM – P070130

31. PROBLEM – P070131

32. PROBLEM – P070132

33. PROBLEM – P070133

34. PROBLEM – P070134

35. PROBLEM – P070135

36. PROBLEM – P070136

37. PROBLEM – P070137

38. PROBLEM – P070138

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