Q1a. Deriving the Electric Potential of a Dipole at a Distant Point

Background

Topic: Electrostatics – Electric Potential of a Dipole

This question tests your understanding of how to derive the electric potential at a point along the axis of a dipole, starting from the potential due to a single point charge, and considering the limit as the distance from the dipole becomes very large.

Key Terms and Formulas

• Electric potential due to a point charge:

• Dipole: Two equal and opposite charges separated by a distance

• For a dipole at large distances:

Step-by-Step Guidance

1. Write the expression for the potential at point due to each charge: and .

2. Add the potentials to get the total potential at : .

3. Combine the terms into a single fraction and simplify the numerator and denominator.

4. For , use a binomial expansion or approximation to simplify the expression further.

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Q1b. Electron Released Near a Dipole: Calculating Speed

Background

Topic: Energy Conservation in Electric Fields

This question tests your ability to use the concept of electric potential energy and conservation of energy to find the speed of an electron moving in the field of a dipole.

Key Terms and Formulas

• Potential energy change:

• Conservation of energy:

• Kinetic energy:

Step-by-Step Guidance

1. Calculate the electric potential at m and m using the dipole formula from part (a).

2. Find the change in electric potential .

3. Determine the change in potential energy for the electron: (where is the elementary charge).

4. Apply energy conservation: The gain in kinetic energy equals the loss in potential energy, .

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Q1b. Calculating Currents in a Multi-Loop Circuit

Background

Topic: DC Circuits – Kirchhoff's Rules

This question tests your ability to analyze a complex circuit using Kirchhoff's laws to determine the current in each resistor.

Key Terms and Formulas

• Kirchhoff's Junction Rule:

• Kirchhoff's Loop Rule: around any closed loop

• Ohm's Law:

Step-by-Step Guidance

1. Label the currents in each branch of the circuit (e.g., , , ).

2. Write Kirchhoff's loop equations for each independent loop in the circuit.

3. Write the junction equation at the node where the branches meet.

4. Set up the system of equations and solve for the currents in the 3 Ω, 1 Ω, 9 Ω, and 18 Ω resistors.

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Q3a. Intensity of Light Through Three Polarizers

Background

Topic: Optics – Polarization of Light

This question tests your understanding of how the intensity of light changes as it passes through multiple polarizers at different angles.

Key Terms and Formulas

• Malus's Law:

• For unpolarized light, first polarizer:

• Subsequent polarizers:

Step-by-Step Guidance

1. Calculate the intensity after the first polarizer: .

2. Calculate the intensity after the second polarizer using Malus's Law: .

3. Calculate the intensity after the third polarizer: (since the angle between the second and third polarizer is ).

4. Express the final intensity as a fraction of .

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