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Chapter 11: Problem 17

According to Wien's displacement law, (A) \(\lambda_{m}=\) constant (B) \(\lambda_{m} T=\) constant (C) \(\lambda_{m} T^{2}=\) constant (D) \(\lambda_{m}^{2} T=\) constant

Short Answer

Expert verified
The correct equation representing Wien's displacement law is (B) \(\lambda_{m} T =\) constant.

Step by step solution

01

Understand Wien's displacement law

Wien's displacement law, named after Wilhelm Wien, states that the wavelength at which the maximum intensity of radiation is emitted by a black body is inversely proportional to its temperature. Mathematically, it can be expressed as: \(\lambda_{m} \propto \frac{1}{T}\) Here, \(\lambda_{m}\) is the wavelength of the maximum intensity and T is the temperature of the object. The constant of proportionality is referred to as Wien's constant (symbol: \(b\)).
02

Express Wien's displacement law as an equation

Wien's displacement law states that the product of the wavelength of the maximum intensity and the temperature is constant (with the constant being Wien's constant). This means that, \(\lambda_{m} T = b\) where b is Wien's constant.
03

Compare the given equations with the Wien's displacement law formula

Now, let's compare the given equations with the formula we derived in step 2 to find the correct equation that represents Wien's displacement law. (A) \(\lambda_{m} =\) constant : This equation states that the wavelength of maximum intensity is a constant. This is incorrect as the wavelength changes with the temperature. (B) \(\lambda_{m} T =\) constant : This equation is exactly the same as the formula derived in step 2, which means that it represents Wien's displacement law correctly. (C) \(\lambda_{m} T^{2} =\) constant : This equation suggests that the wavelength is related to the square of the temperature, which is not true according to Wien's displacement law. (D) \(\lambda_{m}^{2} T =\) constant : This equation implies that the square of the wavelength times the temperature is a constant, which is not the relationship stated in Wien's displacement law.
04

State the correct equation

From the above comparisons, we can see that the correct equation representing Wien's displacement law is: (B) \(\lambda_{m} T =\) constant

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Most popular questions from this chapter

A diatomic molecule having atoms of masses \(m_{1}\) and \(m_{2}\) has its potential energy function about the equilibrium position \(r_{0}\) as given by \(U(r)=-A+B\left(r-r_{0}\right)^{2}\), where \(A\) and \(B\) are constants. When the atom vibrate at high temperature condition, the square of angular frequency of vibration will be (A) \(\frac{2 B}{m_{1}}\) (B) \(\frac{2 B}{m_{2}}\) (C) \(\frac{2 B\left(m_{1}+m_{2}\right)}{m_{1} m_{2}}\) (D) \(\frac{B\left(m_{1}+m_{2}\right)}{2 m_{1} m_{2}}\)

A coil of resistance \(R\) connected to an external battery is placed inside an adiabatic cylinder fitted with a frictionless piston and containing an ideal gas. A current \(I=a_{0} t\) flows through the coil \(\left(a_{0}\right.\) is a teve constant). For time interval \(t=0\) to \(t=t_{0}\), the piston goes up to a height of \((\) Assume \(\Delta \mathrm{U}=0)\) (A) \(\frac{a_{0}^{2} R^{2} t_{0}^{2}}{2 m g}\) (B) \(\frac{a_{0}^{2} R t_{0}^{3}}{2 m g}\) (C) \(\frac{a_{0}^{2} R t_{0}^{3}}{3 m g}\) (D) \(\frac{a_{0}^{2} R t_{0}^{2}}{3 m g}\)

Infrared radiations are detected by (A) spectrometer (B) pyrometer (C) nanometer (D) photometer

It is known that the temperature in the room is \(+20^{\circ} \mathrm{C}\) when the outdoor temperature is \(-20^{\circ} \mathrm{C}\) and \(+10^{\circ} \mathrm{C}\) when the outdoor temperature is \(-40^{\circ} \mathrm{C}\). Then what is the temperature \(T\) of the radiator heating the room? (Assuming that radiated by the heater is proportional to the temperature difference with the room.) (A) \(40^{\circ} \mathrm{C}\) (B) \(60^{\circ} \mathrm{C}\) (C) \(30^{\circ} \mathrm{C}\) (D) \(20^{\circ} \mathrm{C}\)

A block body is at a temperature \(2880 \mathrm{~K}\). The energy radiation emitted by this object with wavelength between \(499 \mathrm{~nm}\) and \(500 \mathrm{~nm}\) is \(U_{1}\), between \(999 \mathrm{~mm}\) and \(1000 \mathrm{~nm}\) is \(U_{2}\), and between \(1499 \mathrm{~nm}\) and \(1500 \mathrm{~nm}\) is \(U_{3}\), then (Wien's constant \(b=2.88 \times 10^{6} \mathrm{~nm}-\mathrm{K}\) ) (A) \(U_{1}>U_{2}\) (B) \(U_{2}>U_{1}\) (C) \(U_{1}=0\) (D) \(U_{3}=0\)
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