Electrostatics

Since the spheres are connected by a conducting wire, the electric potential on both spheres will be the same. The electric field is inversely proportional to the radius, hence the ratio of the electric fields at the surface of the spheres will be 2:1.

When identical liquid drops combine, the potential of the resulting big drop is given by: where is the number of smaller drops. The potential of the big drop is: since 27 drops combine to form 9 larger drops.

The attractive force between two charged spheres is given by Coulomb's law: where: is Coulomb's constant, and are the charges, and is the distance between the charges. The force is greatest when the charges are of opposite signs and the charges are equal, as in the case of and .

The electrical force between two charges is given by Coulomb’s law: where and are the charges, is the distance between them, and is Coulomb's constant.
If the charges are increased and decreased by 10%, we can calculate the new force by considering the change in charges. Let the initial charges be and , and the initial force is .
When one charge is increased by 10% and the other is decreased by 10%, the new charges become and .
The new force is:
Thus, the correct answer is (c).

The force on the positive test charge due to a plane of charge is given by the equation: For the force to be zero, we need to balance the forces on the test charge. Solving the force equation for , we find that the position at which the net force is zero is: Hence, the correct answer is (a).

In a parallel plate capacitor, the charges on the two plates are equal in magnitude but opposite in sign due to the nature of electric charge conservation. One plate gets a positive charge and the other a negative charge. If the total charge on the capacitor is , then the charges on the plates must be and respectively.

Hence, the correct answer is (c).

In a closed organ pipe, the overtones are the harmonics that follow the fundamental frequency. For a closed organ pipe, the harmonic frequencies are odd multiples of the fundamental frequency. If the fundamental frequency is , the frequencies of the first three overtones are:

- First overtone:
- Second overtone:
- Third overtone:

Thus, the ratio of the frequencies of the next three overtones is .

Hence, the correct answer is (d).

Gauss's law states that the electric flux through a closed surface is given by: When the charge is placed at the center of the cube, the total flux is . Since the cube has 6 faces, the flux through each face will be equally distributed. Therefore, the flux through one face is: Hence, the correct answer is (b).

When two spheres are touched together, the charges on them redistribute equally because the spheres are identical. The total charge before they are touched is: After touching and separating, each sphere gets half of the total charge: Hence, the final charge on each sphere is , which corresponds to option (b).

The work done on a charge is given by the equation:

where is the work done, is the charge, is the electric field, and is the displacement. Using the given electric field , the work done is:

Thus, the energy of the charge at is given by , where is the displacement. This represents the total energy transferred to the charge as it moves in the electric field.

Hence, the correct answer is (c).

When a metal sheet is inserted between the plates of a parallel plate capacitor, it divides the capacitor into two separate capacitors. The sheet being a conductor will experience equal and opposite charges on its faces, thus maintaining charge balance. The battery does not supply more charge because the overall potential difference remains the same. The capacitance does not increase; it is affected by the properties of the dielectric, but the metal sheet acts as a conductor and does not serve as a dielectric.

Hence, the correct answer is (d).

The fundamental frequency of a pipe is inversely proportional to its length. For a pipe with both ends open, the fundamental frequency is given by: where is the velocity of sound and is the length of the pipe. The shorter the pipe, the higher the frequency. Among the options, the 1.6 m pipe will have the highest fundamental frequency. Hence, the correct answer is (c).

In the given setup, the electron, which is negatively charged, will experience an attractive force toward the positively charged objects. The force on the electron depends on the distribution of the charges. The net force is directed toward the corner where the resultant of the forces from the surrounding charges is greatest. From the diagram, the net force will be directed toward point D. Hence, the correct answer is (d).