(b) (i) Discuss briefly diffraction of light from a single slit and draw the shape of the diffraction pattern. (ii) An object is placed between the pole and the focus of a concave mirror. Using the mirror formula, prove mathematically that it produces a virtual and an enlarged image.
Solution
(i) A plane wavefront on a single slit (width a) produces secondary wavelets (Huygens). Along the axis they reinforce to give a bright central maximum; for direction θ the path difference asinθ gives minima at asinθ=nλ and weak secondary maxima at asinθ=(2n+1)2λ. The intensity shape is a dominant central band (twice as wide) flanked by rapidly decreasing side bands (see figure). (ii) For a concave mirror take f negative; an object between pole and focus has ∣u∣<∣f∣. From the mirror formula
v1+u1=f1⇒v=u−ffu
Since ∣u∣<∣f∣, u−f has the sign that makes v positive, i.e. the image is formed behind the mirror — it is virtual. The magnification is
m=−uv=f−uf=1+f−uu
with ∣u∣<∣f∣ this gives ∣m∣>1, so the image is enlarged (and erect).