A7 Let be the value of for which is maximum. · OlympiadBench #1066

Physics · modern-physics

Problem

A7 Let

x_{0}

be the value of

x_{\mathrm{g}}

for which

\eta

is maximum. Obtain the cubic equation that gives

x_{0}

. Estimate the value of

x_{0}

within an accuracy of \pm 0.25. Hence calculate

\eta\left(x_{0}\right)

. Given constants: Particles from the Sun

{ }^{1}

Photons from the surface of the Sun and neutrinos from its core can tell us about solar temperatures and also confirm that the Sun shines because of nuclear reactions. Throughout this problem, take the mass of the Sun to be

M_{\odot}=2.00 \times 10^{30} \mathrm{~kg}

, its radius,

R_{\odot}=7.00 \times

10^{8} \mathrm{~m}

, its luminosity (radiation energy emitted per unit time),

L_{\odot}=3.85 \times 10^{26} \mathrm{~W}

, and the Earth-Sun distance,

d_{\odot}=1.50 \times 10^{11} \mathrm{~m}

. Note: (i)

\int x e^{a x} d x=\left(\frac{x}{a}-\frac{1}{a^{2}}\right) e^{a x}+

constant (ii)

\int x^{2} e^{a x} d x=\left(\frac{x^{2}}{a}-\frac{2 x}{a^{2}}+\frac{2}{a^{3}}\right) e^{a x}+

constant (iii)

\int x^{3} e^{a x} d x=\left(\frac{x^{3}}{a}-\frac{3 x^{2}}{a^{2}}+\frac{6 x}{a^{3}}-\frac{6}{a^{4}}\right) e^{a x}+

constant A Radiation from the sun : Context question: A.1 Assume that the Sun radiates like a perfect blackbody. Use this fact to calculate the temperature,

T_{\mathrm{s}}

, of the solar surface. Context answer:

5.76 \times 10^{3}

Extra Supplementary Reading Materials: The spectrum of solar radiation can be approximated well by the Wien distribution law. Accordingly, the solar energy incident on any surface on the Earth per unit time per unit frequency interval,

u(v)

, is given by

u(v)=A \frac{R_{\odot}^{2}}{d_{\odot}^{2}} \frac{2 \pi h}{c^{2}} v^{3} \exp \left(-h v / k_{\mathrm{B}} T_{\mathrm{s}}\right)

where

v

is the frequency and

A

is the area of the surface normal to the direction of the incident radiation. Now, consider a solar cell which consists of a thin disc of semiconducting material of area,

A

, placed perpendicular to the direction of the Sun's rays. Context question: A2 Using the Wien approximation, express the total radiated solar power,

P_{\mathrm{in}}

, incident on the surface of the solar cell, in terms of

A, R_{\odot}, d_{\odot}, T_{\mathrm{S}}

and the fundamental constants

c, h, k_{\mathrm{B}}

Context answer:

P_{\mathrm{in}}=\frac{12 \pi k_{\mathrm{B}}^{4}}{c^{2} h^{3}} T_{\mathrm{s}}^{4} A \frac{R_{\odot}^{2}}{d_{\odot}^{2}}

Context question: A3 Express the number of photons,

n_{\gamma}(v)

, per unit time per unit frequency interval incident on the surface of the solar cell in terms of

A, R_{\odot}, d_{\odot}, T_{\mathrm{s}}, v

and the fundamental constants

c, h, k_{\mathrm{B}}

. Context answer:

n_{\gamma}(\nu)=A \frac{R_{\odot}^{2}}{d_{\odot}^{2}} \frac{2 \pi}{c^{2}} \nu^{2} e^ {\frac{-h \nu }{k_{\mathrm{B}} T_{\mathrm{s}}}}

Extra Supplementary Reading Materials: The semiconducting material of the solar cell has a "band gap" of energy,

E_{\mathrm{g}}

. We assume the following model. Every photon of energy

E \geq E_{\mathrm{g}}

excites an electron across the band gap. This electron contributes an energy,

E_{\mathrm{g}}

, as the useful output energy, and any extra energy is dissipated as heat (not converted to useful energy). Context question: A4 Define

x_{\mathrm{g}}=h v_{\mathrm{g}} / k_{\mathrm{B}} T_{\mathrm{s}}

where

E_{\mathrm{g}}=h v_{\mathrm{g}}

. Express the useful output power of the cell,

P_{\text {out }}

, in terms of

x_{\mathrm{g}}, A

R_{\odot}, d_{\odot}, T_{\mathrm{s}}

and the fundamental constants

c, h, k_{\mathrm{B}}

. Context answer:

P_{\text {out }}=\frac{2 \pi k_{\mathrm{B}}^{4}}{c^{2} h^{3}} T_{\mathrm{s}}^{4} A \frac{R_{\odot}^{2}}{d_{\odot}^{2}} x_{\mathrm{g}}(x_{\mathrm{g}}^{2}+2 x_{\mathrm{g}}+2) e^{-x_{\mathrm{g}}}

Context question: A5 Express the efficiency,

\eta

, of this solar cell in terms of

x_{\mathrm{g}}

. Context answer:

\eta=\frac{x_{\mathrm{g}}}{6}\left(x_{\mathrm{g}}^{2}+2 x_{\mathrm{g}}+2\right) e^{-x_{\mathrm{g}}}

Context question: A6 Make a qualitative sketch of

\eta

versus

x_{\mathrm{g}}

. The values at

x_{\mathrm{g}}=0

and

x_{\mathrm{g}} \rightarrow \infty

should be clearly shown. What is the slope of

\eta\left(x_{\mathrm{g}}\right)

x_{\mathrm{g}}=0

and

x_{\mathrm{g}} \rightarrow \infty

? Context answer:

\frac{1}{3}

, 0

Intuitive explanation

To find the maximum efficiency, you need to recognize that $\eta$ is a function of the dimensionless band-gap parameter $x_g$ , and maxima occur where the derivative equals zero. Taking $d\eta/dx_g = 0$ will yield a cubic equation in $x_0$ (the optimal band gap). This is a classic optimization problem: the interplay between collecting more photons (smaller $x_g$ ) and extracting more useful energy per photon (larger $x_g$ ) produces a sweet spot that you can find algebraically and then refine numerically.

Intuitive explanation

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