To find the distance \(d\) over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density \(\rho\) of the fog, intensity (power/area) \(S\) of the light from the signal and its frequency \(f\). The engineer finds that \(d\) isproportionalto \(S^{1 / n}\). The value of \(n\) is

JEE Advanced · 2014 · 25 May 2014 · Paper 1 · Units & Measurement

To find the distance

d

over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density

ρ

of the fog, intensity (power/area)

S

of the light from the signal and its frequency

f

. The engineer finds that

d

isproportionalto

S^{1/ n}

. The value of

n

Answer: 3

Solution

Assume

d = ρ^{a} S^{b} f^{c}

.

Dimensions:

[d] = [L]

[ρ] = [M L^{- 3}]

[S] = [M T^{- 3}]

[f] = [T^{- 1}]

.

So

[L] = [M L^{- 3}]^{a} [M T^{- 3}]^{b} [T^{- 1}]^{c} = [M^{a + b} L^{- 3 a} T^{- 3 b - c}]

.

Equating exponents:
For M:

a + b = 0

For L:

- 3 a = 1

, so

a = - \frac{1}{3}

For T:

- 3 b - c = 0

From

a + b = 0

b = \frac{1}{3}

.
From

- 3 b - c = 0

c = - 3 b = - 1

.

Thus

d \propto ρ^{- 1/3} S^{1/3} f^{- 1}

, so

d \propto S^{1/3}

, giving

n = 3

JEE Advanced · 2014 · 25 May 2014 · Paper 1 · Integer answer

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JEE Advanced · 2014 · 25 May 2014 · Paper 1 · Units & Measurement

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