Electrostatics

Step 1: Understanding the potential formula.
The electric potential at a point due to a point charge is given by: where is Coulomb's constant, is the charge, and is the distance from the charge. The total potential at point A will be the sum of the potentials due to both charges.
Step 2: Applying the formula for multiple charges.
Using the distances from point A to each of the charges and substituting the appropriate values, the total potential at point A is calculated as:
Step 3: Conclusion.
The correct potential at point A is , so the correct answer is (A).

Step 1: Electric potential formula.
The potential across a resistor is related to the charge and the resistance by the formula: where is the capacitance of the system.
Step 2: Using the charge.
Since the total charge is , the potential across the resistor will be directly proportional to .
Step 3: Conclusion.
Thus, the potential will depend on the value of , so the correct answer is (C).

Step 1: Understanding the electric field.
The electric field between the plates of a capacitor is given by the formula: where: - is the voltage across the plates, - is the distance between the plates.
Step 2: Applying the change in voltage.
If the capacitor is connected to a battery with a voltage across the plates, and the charge on the capacitor is , the electric field will double.
Step 3: Conclusion.
The electric field will become , so the correct answer is (C).

Step 1: Understanding the potential formula.
The electric potential at a point due to a point charge is given by: where is Coulomb's constant, is the charge, and is the distance from the charge.
Step 2: Applying the potential for a spherical shell.
For a spherical shell, the potential at a point is determined by the charge at the center and the distance from the center. Given that the potential due to the shell is equivalent to the potential due to a point charge at the center, the result will be .
Step 3: Conclusion.
Thus, the correct answer is (B) .

Step 1: Understanding electric potential.
The electric potential at a point in space is the same for both charges when the system is symmetrical and the charges are placed in an evenly distributed electric field.
Step 2: Conclusion.
Since the electric field is applied equally to both charges, the electric potential at both points will be the same, hence the correct answer is (A).

Step 1: Understanding the Concept:
The potential energy of a dipole in an electric field is . The work done in rotating the dipole is the change in its potential energy: .
Step 2: Detailed Explanation:
Initial angle ( ) = 0° (parallel to the field). Final angle ( ) = 90°. $ PE$.

Step 1: Understanding the Concept:
For a charged spherical shell, the electric field inside is zero. This means the electric potential remains constant at all points inside and is equal to the potential on the surface.
Step 2: Detailed Explanation:
The potential on the surface ( ) is . Since the potential is constant inside the shell ( ), the potential at is exactly the same as the potential on the surface. $ $
Step 3: Final Answer:
The correct option is (a).

Step 1: Understanding the Concept:
The capacitance of a parallel plate capacitor is , where is the dielectric constant and is the distance.
Step 2: Detailed Explanation:
Initial state (Air, ): .
Final state (Dielectric , Distance ): .
Substitute into the second equation: $ $
Step 3: Final Answer:
The dielectric constant is 4.0.

Step 1: Understanding the Concept:
The electric field due to an infinitely long wire at a distance is . We must use superposition to find the net field at the midpoint.
Step 2: Detailed Explanation:
The midpoint is at a distance from each wire. 1. Field due to : (directed away from the positive wire). 2. Field due to : (directed toward the negative wire). At the midpoint, both fields point in the same direction (from positive to negative). $ $
Step 3: Final Answer:
The correct option is (a).

Step 1: Understanding the Concept:
According to Gauss's Law, the total electric flux through a closed surface is times the net charge enclosed. If a charge is not fully enclosed, we must imagine a symmetrical structure to enclose it.
Step 2: Detailed Explanation:
A charge placed at the corner (vertex) of a cube is shared by 8 identical cubes meeting at that point. To fully enclose the charge symmetrically, we need a larger "super-cube" made of 8 smaller cubes. The total flux through this large cube is . Since the flux is distributed equally among the 8 cubes, the flux through one single cube is: $ $
Step 3: Final Answer:
The correct option is (b).

Step 1: Understanding the Concept:
Force in an electric field is . According to Newton's Second Law, . Therefore, acceleration .
Step 2: Detailed Explanation:
For a Proton ( ): Charge , Mass .
For an Alpha particle ( ): Charge , Mass .
$ \frac{a_p}{a_\alpha} = \frac{eE/m}{eE/2m} = \frac{2}{1}$.
Step 3: Final Answer:
The correct option is (c).