Q1. Electrostatics and Capacitance

Background

Topic: Electrostatics, Gauss' Law, Electric Field, Electric Potential, Capacitance

This question tests your understanding of charge transfer, electric fields from charged spheres, dipole fields, and capacitor circuits.

Key Terms and Formulas

• Gauss' Law:

• Electric field outside a charged sphere:

• Dipole field (for ): , where

• Capacitance in series and parallel: ,

• Charge stored:

Step-by-Step Guidance

1. For part (a), consider how to transfer charge using a negatively charged object and an insulated metal sphere. Think about the process of grounding and contact.

2. For part (b), use Gauss' Law to derive the electric field outside a conducting sphere. Set up a Gaussian surface at radius and apply the law.

3. For part (c), analyze the diagram (see below) and use the result from part (b) to find the field at point P due to both spheres. Express the field in terms of , , and for .

4. For part (f), examine the capacitor circuit (see below). Identify which capacitors are in series and which are in parallel, then write the formula for equivalent capacitance and total charge.

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Q2. Magnetism and Electromagnetism

Background

Topic: Magnetic Force, Gauss's Law for Electricity and Magnetism, Forces Between Wires, Magnetic Field from Current Loops

This question tests your understanding of the direction of magnetic fields, Gauss's Law for magnetism, and calculation of forces and fields in wire configurations.

Key Terms and Formulas

• Magnetic force:

• Gauss's Law for electricity:

• Gauss's Law for magnetism:

• Force between parallel wires:

• Magnetic field at center of arc:

Step-by-Step Guidance

1. For part (a), use the right-hand rule to determine the direction of the magnetic field based on the force direction.

2. For part (b), write both forms of Gauss's Law and explain why magnetic flux through a closed surface is always zero.

3. For part (c), set up the formula for the force between two parallel wires, define all variables, and sketch the vectors.

4. For part (d), use the diagram (see below) and apply the formula for the magnetic field at point P due to two arcs. Consider the direction (into or out of the page).

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Q3. Optics and AC Circuits

Background

Topic: Refraction, Total Internal Reflection, AC Current, LRC Circuits

This question tests your understanding of the index of refraction, critical angle, AC current properties, and damped oscillations in circuits.

Key Terms and Formulas

• Index of refraction:

• Critical angle:

• AC current:

• Average current:

• Mean square current:

• Current amplitude:

• Damped angular frequency:

Step-by-Step Guidance

1. For part (a), use the change in wavelength to calculate the index of refraction. Set up the formula and substitute values.

2. For part (a)(ii), use the index to find the critical angle for total internal reflection.

3. For part (b), recall the formulas for average current, mean square current, and amplitude in AC circuits.

4. For part (c), define all variables in the damped frequency formula and sketch the charge vs. time for underdamped oscillations.

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Q4. Optics: Lenses, Mirrors, and Interference

Background

Topic: Lens Equation, Magnification, Ray Diagrams, Interference Patterns

This question tests your understanding of image formation by lenses and mirrors, ray tracing, and interference minima.

Key Terms and Formulas

• Lens equation:

• Magnification:

• Interference minima:

Step-by-Step Guidance

1. For part (a), use the lens equation to find the image distance. Substitute cm, cm.

2. Calculate magnification using the formula and the values found in the previous step.

3. For part (c), set up the formula for interference minima and substitute m, m.

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