N lamps each of resistance r, are fed by the machine of resistance R. If light emitted by any lamp is proportional to the square of the heat produced, prove that the most efficient way of arranging them is to place them in parallel arcs, each containing n lamps, where n is the integer nearest to.

${(\frac{r}{NR})}^{3 / 2}$

${(\frac{NR}{r})}^{1 / 2}$

(NRr)^3/2

(NRr)^1/2

Solution

${(\frac{NR}{r})}^{1 / 2}$

As each is containing n lamps, hence,

The resistance of each arc = nr

Number of arcs = N/n

Total resistance S is given by

$\frac{1}{S} = Σ \frac{1}{nr} = \frac{N}{n} (\frac{1}{nr}) S = \frac{n^{2} r}{N} ∴ Total resistance = R + S + (n^{2} r) N$

If E is the emf of the machine, current entering the arcs is E/(R+S) and in each arc is nE/(R+S)N.

Hence, the current passing through each lamp.

$I = \frac{nE}{N (R + n^{2} r / N)} = \frac{E}{N} {[\frac{R}{n} + \frac{nr}{N}]}^{- 1}$

Now heat produced per second in the lamps is

H = NI²

since, light emitted is proportional to H2 therefore, the light produced is maximum when H² and hence H is maximum or $[\frac{R}{n} + \frac{nr}{N}]$ is minimum. Hence, we can write,

$\frac{R}{n} + \frac{nr}{N} = [{(\frac{R}{n})}^{1 / 2} - {(\frac{nr}{N})}^{1 / 2}] + 2 {(\frac{Rr}{N})}^{1 / 2}$

This is minimum when $\sqrt{R / n} - \sqrt{nr / N} = 0$ or very small or n is closely equal to (NR/r)^1/2

Some More Questions From Nuclei Chapter

Name the electromagnetic radiations used for (a) water purification, and (b) eye surgery

The above is a plot of binding energy per nucleon Eb, against the nuclear mass M; A, B, C, D, E, F correspond to different nuclei. Consider four reactions:

(i) A + B → C + ε (ii) C → A + B + ε
(iii) D + E → F + ε and (iv) F → D + E +ε

where ε is the energy released? In which reaction is ε positive?

This question contains Statement -1 and Statement-2. Of the four choices given after the statements,
Choose the one that best describes the two statements.
Statement – I: Energy is released when heavy nuclei undergo fission or light nuclei undergo fusion.
and
Statement – II: For heavy nuclei, binding energy per nucleon increases with increasing Z while for light nuclei it decreases with increasing Z.

If M₀ is the mass of an oxygen isotope ₈O¹⁷ , Mp and M_N are the masses of a proton and a neutron respectively, the nuclear binding energy of the isotope is

For Daily Free Study Material Join wiredfaculty

Nuclei

Some More Questions From Nuclei Chapter

Name the electromagnetic radiations used for (a) water purification, and (b) eye surgery

If M₀ is the mass of an oxygen isotope ₈O¹⁷ , Mp and M_N are the masses of a proton and a neutron respectively, the nuclear binding energy of the isotope is

In gamma-ray emission from a nucleus

When ₃Li⁷ nuclei are bombarded by protons, and the resultant nuclei are ₄Be⁸ , the emitted particles will be

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction

For Daily Free Study Material Join wiredfaculty

Nuclei

Some More Questions From Nuclei Chapter

Name the electromagnetic radiations used for (a) water purification, and (b) eye surgery

If M0 is the mass of an oxygen isotope 8O17 , Mp and MN are the masses of a proton and a neutron respectively, the nuclear binding energy of the isotope is

In gamma-ray emission from a nucleus

When 3Li7 nuclei are bombarded by protons, and the resultant nuclei are 4Be8 , the emitted particles will be

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

If M₀ is the mass of an oxygen isotope ₈O¹⁷ , Mp and M_N are the masses of a proton and a neutron respectively, the nuclear binding energy of the isotope is

When ₃Li⁷ nuclei are bombarded by protons, and the resultant nuclei are ₄Be⁸ , the emitted particles will be