2011 Semifinal Exam Part A 4 Question A3 A light bulb has a solid cylindrical filament of length L and radius a,and consumes power P. You are to design a new light bulb,using a cylindrical filament of the same material,operating at the same voltage,and emitting the same spectrum of light,which will consume power nP.What are the length and radius of the new filament?Assume that the temperature of the filament is approximately uniform across its cross-section;the filament doesn't emit light from the cnds:and energy loss due to convection is minimal. Question A4 In this problem we consider a simplified model of the electromagnetic radiation inside a cubical box of side length L.In this model,the electric field has spatial dependence E(,y,2)=Eosin(kz)sin(kuy)sin(kz2) where one corner of the box lies at the origin and the box is aligned with the a,y,and z axes.Let h be Planck's constant,kB be Boltzmann's constant,and c be the speed of light. a.The electric field must be zero everywhere at the sides of the box.What condition does this impose on k,ky,and k2?(Assume that any of these may be negative,and include cases where one or more of the ki is zero,even though this causes E to be zero.) b.In the model,each permitted value of the triple (k,ky,k)corresponds to a quantum state. These states can be visualized in a state space,which is a notional three-dimensional space with axes corresponding to kz,ky,and kz.How many states occupy a volume s of state space, if s is large enough that the discreteness of the states can be ignored? c.Each quantum state,in turn,may be occupied by photons with frequencyw-ck, f where kl=Vka2+ky2+kz2 In the model,if the temperature inside the box is T,no photon may have energy greater than kBT.What is the shape of the region in state space corresponding to occupied states? d.As a final approximation,assume that each occupied state contains exactly one photon.What is the total energy of the photons in the box,in terms of h,kB,c,T,and the volume of the box V?Again,assume that the temperature is high enough that there are a very large number of occupied states.(Hint:divide state space into thin regions corresponding to photons of the same energy. Note that while many details of this model are extremely inaccurate,the final result is correct except for a numerical factor. Copyright C2011 American Association of Physics Teachers