1. (AC Circuits). Consider an LC circuit (in series) with a driving emf ε = ε_max cosωt.

a) Show that if ω >> √(LC), the amplitude of the potential difference across the terminals is ΔV ≡ (X_C/X_L) ε_max.

b) Show that ΔV is much smaller than ε_max. Thus, the circuit can be used as a filter that strongly attenuates high-frequency components in the driving emf (that is, a low-pass filter).

2. (Electric Field of Accelerated Charge). We have seen that accelerating charges can produce electromagnetic waves: electric and magnetic fields that spread out from the position of the charge like ripples on a pond into which a stone has been dropped

When the charge accelerates, the spherical electric field gets "compressed" in the direction of the motion. This establishes a "kink" in the normally radial field. Hence, a transverse component of the electric field arises. This transverse component is called a radiation field. It looks like:

E_θ =k(qa sinθ/c²r)

where a = v/τ  is the charge's acceleration (we assume that the charge accelerates only for a short time-interval τ).

In a collision with an atom, an electron suffers a deceleration of 2.0 x 10²⁴ m/s². What is the magnitude of the electric radiation field generated by this electron at a distance of 20 cm at an angle of 45° in the direction of the deceleration? At what time after the instant of collision does this radiation field arrive at that distance?

3. (Electromagnetic Flu; Intensity). At night, the naked, dark-adapted eye can see a star provided the energy flux reaching the eye is 8.8 x 10^-11 W/m².

a) Under these conditions, how many watts of power enter the eye? The diameter of the dark-adapted pupil is 7.0 mm.

b) Assume that in our neighborhood, there are, on average 3.5 x 10^-5 stars per cubic light-year and that each of these emits the same amount of light as the Sun (3.9 x 10²⁶ W). If so, how many stars could we see in the sky with the naked eye? How far would the faintest visible star be?

4. (Refraction, Thin Lenses). Show that if a lens of index of refraction n is placed in a medium (e.g., water) of index of refraction n', then the lens-maker's equation becomes

1/f = ((n/n')-1)(1/R₁ + 1/R₂)

5. (Interference, Thin Films) White light is incident normally on a thin soap film (n = 1.33) suspended in air.

a) What are the two minimum thicknesses that will constructively reflect yellow (λ = 590 nm) light?

b) What are the two minimum thicknesses that will destructively reflect yellow light?