1. (AC Circuits). Consider an LC circuit (in series) with a driving emf ε = εmax cosωt.
a) Show that if ω >> √(LC), the amplitude of the potential difference across the terminals is ΔV ≡ (XC/XL) εmax.
b) Show that ΔV is much smaller than εmax. 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:
Eθ =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?