Question

The intensity produced by a long cylindrical light source at a small distance r from the source is proportional to
(a) 1r2
(b) 1r3
(c) 1r
(d) none of these

Answer 1

Explanation:

which is correct option number d

Answer 2

The intensity produced by a long cylindrical light source at a small distance r from the source is proportional to $\frac{1}{r}$

Explanation:

Step 1:

Let's find two cylindrical coaxial surfaces at distances r and r ' from the axis. Let dA and dA ' subtend at the central axis the solid angle d. The height of the area dimension is going to be the same, that is, equal to dy.  

Let the breath of dA be dx and that of dA' be dx'.  

Of the arcs now,

$d x=r d \theta$

$d x^{\prime}=r^{\prime} d \theta$

$d A=d x d y=r d \theta d y$

$d A^{\prime}=d x^{\prime} d y=r^{\prime} d \theta d y$

$\frac{d A}{d A^{\prime}}=\frac{r}{r^{\prime}}$

$\frac{d A}{r}=\frac{d A^{\prime}}{r^{\prime}}=d \omega$

Step 2:

The luminous flux of the strong angle d below will be:

$d F=I d \omega$

Now,  

$d F=I \frac{d A}{r}$

Step 3:

If the surfaces are α-angle oriented,

$d F=I \frac{d A \cos \alpha}{r}$

Now illumination is known as  

$E=\frac{d F}{d A}=I \frac{d A \cos \alpha}{r}$

$E \propto \frac{1}{r}$

Therefore the correct answer is Option (c)  $\frac{1}{r}$