Principle of Superposition

Q. Three charges each of magnitude q are placed at the corners of an equilateral triangle, the electrostatic force on the charge placed at the center is ( each side of triangle is L)

1. Zero
2. $\frac{1}{4\pi {ϵ}_o}\frac{q^2}{L^2}$
   
3. $\frac{1}{4\pi {ϵ}_o}\frac{3q^2}{L^2}$
   
4. $\frac{1}{12\pi {ϵ}_o}\frac{q^2}{L^2}$

Q. ABC is a right angled triangle in which AB=3 cm and BC=4 cm . And $\angle ABC=\frac{\pi }{2}$. The three charges +15, +12 and -20 e.s.u. are placed respectively on A, B and C. The force acting on B is

1. 125 dynes
2. 35 dynes
3. 25 dynes
4. Zero

Q.

The distance between the two identical charges is reduced by$\frac{1}{4}$. The force of repulsion between them is

1. $\frac{1}{4}$of initial force

2. $4$ times the initial force

3. $16$ times the initial force

4. $8$ times the initial force

Q. Four positive charges $(2\sqrt{2}-1)Q$ are arranged at the four corners of a square. Another charge $q$ is placed at the centre of the square. For what value of $q$ , is the resulting force acting on each corner charge be zero ?

1. $-Q$
2. $-(\sqrt{2}+1)Q$
3. $\frac{-4Q}{7}$
4. $\frac{-7Q}{4}$

Q. Two pulses of identical shape overlap such that the displacement of the rope is momentarily zero at all points, what happens to the energy at this time.

1. Energy becomes zero
2. The total energy is in form of potential energy
3. The total energy is in form of kinetic energy
4. None of these

Q. A unit positive charge has to be brought from infinity to a mid-point between two charges, $20\mu \text{C}$ and $10\mu \text{C}$ separated by a distance of $50\text{m}$. How much work will be required by an external force?

1. $2.1\times {10}^4\text{J}$
2. $1.1\times {10}^4\text{J}$
3. $3.1\times {10}^4\text{J}$
4. $0.1\times {10}^4\text{J}$

Q. Two similar spheres having +q and -q charge are kept at a certain distance. F force acts between the two. If in the middle of two spheres, another similar sphere having +q charge is kept, then it experiences a force whose magnitude and direction are

1. Zero having no direction
2. 8F towards +q charge
3. 8F towards -q charge
4. 4F towards +q charge

Q. A charge $q_1$ exerts some force on a second charge $q_2$. If third charge $q_3$ is brought near, the force of $q_1$ exerted on $q_2$

1. Decreases
2. Increases
3. Remains unchanged
4. Increases if $q_3$ is of the same sign as $q_1$ and decreases if $q_2$ is of opposite sign

Q. A charge ${10}^{-9}\text{C}$ is located at the origin of a coordinate system and another charge $Q$ at $(2,0,0)$. If the $x-$ component of the electric field at $(3,1,1)$ is zero, calculate the value of $Q$.

1. $4.3\times {10}^{-10}\text{C}$
2. $-2.3\times {10}^{-10}\text{C}$
3. $-4.3\times {10}^{-10}\text{C}$
4. $2.3\times {10}^{-10}\text{C}$

Q. A charge $q_1$ exerts some force on a second charge $q_2$. If third charge $q_3$ is brought near, the force of $q_1$ exerted on $q_2$

1. Decreases
2. Increases
3. Remains unchanged
4. Increases if $q_3$ is of the same sign as $q_1$ and decreases if $q_2$ is of opposite sign

Q. Three equal negative charges, $-q_1$ each, form the vertices of an equilateral triangle. A particle of mass $m$ and a positive charge $q_2$ is constrained to move along a line perpendicular to the plane of triangle and through its centre which is at a distance $r$ from each of the negative charges as shown in the figure. The whole system is kept in gravity free space. Find the time period of vibration of the particle for small displacement from equilibrium position.

1. $2\pi \sqrt{\frac{4\pi {ϵ}_{0}m{r}^3}{q_1{q}_2}}$
2. $2\pi \sqrt{\frac{3\pi {ϵ}_{0}m{r}^3}{4q_1{q}_2}}$
3. $2\pi \sqrt{\frac{4\pi {ϵ}_{0}m{r}^3}{3q_1{q}_2}}$
4. $2\pi \sqrt{\frac{\pi {ϵ}_{0}m{r}^3}{3q_1{q}_2}}$

Q. Two waves represented by equation,
$y_1=2\sin (\pi t-0.1x)$ and $y_2=2\sin (\pi t-0.1x+\varphi )$ are superimposed. If the resultant wave has a maximum amplitude equal to $4\text{cm}$, the value of $\varphi$ is :

1. $\pi$
2. $2\pi$
3. $\frac{\pi }{2}$
4. $\frac{3\pi }{2}$

Q. Five point charges, $+q$ each, are placed at the five vertices of a regular hexagon. The distance of centre of the hexagon from any of the vertices is $a$. The electric field at the centre of the hexagon is

1. $\frac{q}{16\pi {ϵ}_0{a}^2}$
2. Zero
3. $\frac{q}{8\pi {ϵ}_0{a}^2}$
4. $\frac{q}{4\pi {ϵ}_0{a}^2}$

Q. Two coherent sources, emitting waves of wavelength $\lambda$, are placed at a separation of $3\lambda$. A detector is moving around them on a large circle. The number of minima detected by it in one complete rotation is

1. $2$
2. $4$
3. $8$
4. $12$

Q. A point negative charge $-Q$ is placed at a distance $r$ from a dipole with dipole moment $P$ in the $\text{x-y}$ plane as shown in figure

The $\text{x-}$component of force acting on the charge $-Q$ is

1. $-\frac{2PkQ}{r^3}\cos \theta \hat{i}$

2. $\frac{2PkQ}{r^3}\cos \theta \hat{i}$

3. $\frac{PkQ}{r}\cos \theta \hat{i}$

4. $-\frac{PkQ}{r}\cos \theta \hat{i}$

Q. Two interfering waves have the same wavelength, frequency and amplitude. They are travelling in the same direction but ${90}^{\circ }$ out of phase compared to individual waves. The resultant wave will have the same

1. amplitude and velocity but different wavelength
2. frequency and velocity but different wavelength
3. wavelength and velocity but different amplitude
4. amplitude and frequency but different wavelength

Q. Six charges, three positive and three negative of equal magnitude are to be placed at the vertices of a regular hexagon such that the electric field at O is double the electric field when only one positive charge of same magnitude is placed at R. Which of the following arrangements of charges is possible for P, Q, R, S, T and U respectively

1. +, -, +, -, -, +
2. +, -, +, -, +, -
3. +, +, -, +, -, -
4. -, +, +, -, +, -

Q. Two pulses are propagating on a stretched string as shown in figure. The plane of propagation of pulses are perpendicular to each other. When the two pulses overlaps the resultant displacement of any particle is

1. Algebraic sum of the individual displacement due to each wave pulse
2. Vector sum of the individual displacement due to each wave pulse
3. Independent of the individual displacement due to each wave pulse.
4. None of these

Q. A large spherical mass $M$ is fixed at one position and two identical point masses $m$ are kept on a line passing through the centre of $M$ as shown in the figure.The point masses are connected by a rigid massless rod of length $l$ and this assembly is free to move along the line connecting them. All three masses interact only through their mutual gravitational interaction.When the point mass nearer to $M$ is at distance $r=3l$ from $M$, the tension in the rod is zero for $m=k\left(\frac{M}{288}\right)$. The value of $k$ is

Q. Two monochromatic light waves of amplitudes $A$ and $2A$ interfering at a point, have a phase differences of ${60}^{\circ }$. The intensity of the resultant wave at that point will be proportional to

1. $9A^2$
2. $3A^2$
3. $5A^2$
4. $7A^2$

Q. Two identical pulses whose centres are initially $16\text{cm}$ apart are moving towards each other on a stretched string with same speed as shown in the figure. After $4$ seconds, if the vertical displacement for resultant pulse is zero then the speed of pulses are

1. $2\text{cm/s}$
2. $1\text{cm/s}$
3. $4\text{cm/s}$
4. $8\text{cm/s}$

Q. Two charges of $40\mu C$ and $90\mu C$ are placed at a distance of $1\text{m}$ from each other. Where should a third charge be placed on the line joining them, if total force on the third charge is zero?

1. In between, $40\text{cm}$ from $90\mu C$
2. In between, $40\text{cm}$ from $40\mu C$
3. In between, $50\text{cm}$ from $40\mu C$
4. In between, $20\text{cm}$ from $90\mu C$

Q. Two waves generated from incoherent sources, are propagating on a stretched string.

Statement $1$: The resultant wave can be obtained by using the principle of superposition of waves.
Statement $2$: Principle of superposition of waves is not applicable on the wave obtained from incoherent sources.

1. Statement $1$ is true, statement $2$ is false
2. Statement $1$ is false, statement $2$ is true
3. Both the statements are true.
4. Both the statements are false.

Q. Lenz's law gives

1. The magnitude of the induced e.m.f.
2. The direction of the induced current
3. Both the magnitude and direction of the induced current
4. The magnitude of the induced current

Q.

The distance of a point from the earth's centre where the resultant gravitational field due to the earth and the moon is zero is...( The mass of the earth is 6.0$\times$ ${10}^{24}$ kg and that of the moon is 7.4$\times$ ${10}^{22}$ kg. The distance between the earth and the moon is 40 $\times$ ${10}^5$ km.

1. $1.6\times {10}^{5}km$

2. $3.6\times {10}^{5}km$

3. $4.6\times {10}^{5}km$

4. $6\times {10}^{5}km$