While conducting the Young's double slit experiment, a student replaced the two slits with a large opaque plate in the XY-plane containing two small holes that act as two coherent point sources (S1, S2) emitting light of wavelength 600 mm. The student mistakenly placed the screen parallel to the XZ-plane (for z > 0) at a distance D = 3 m from the mid-point of S1S2, as shown schematically in the figure. The distance between the source d = 0.6003 mm. The origin O is at the intersection of the screen and the line joining S1S2. Which of the following is(are) true of the intensity pattern on the screen?

Hyperbolic bright and dark bands with foci symmetrically placed about O in the x-direction

Semi circular bright and dark bands centered at point O

The region very close to the point O will be dark

Straight bright and dark bands parallel to the X-axis.

Wave Optics PYQ — JEE Advanced 2016 · Shift 2 · Q48 with Solution

View written solutionFree

Correct answer: D

1. Geometry of the setup

Let the two coherent point sources be at $S_1\left(-\frac d2,0,0\right), \qquad S_2\left(+\frac d2,0,0\right)$ in the $XY$ -plane.

The screen is parallel to the $XZ$ -plane, so its equation is $y=D$ with $D=3\,\text{m}$ .

A general point on the screen is $P(x,D,z).$

The origin $O$ is the point where the line joining $S_1S_2$ meets the screen, so $O=(0,D,0).$

2. Distances from the two sources to a point on screen

The distances are $r_1=\sqrt{\left(x+\frac d2\right)^2+D^2+z^2},$ $r_2=\sqrt{\left(x-\frac d2\right)^2+D^2+z^2}.$

Hence the path difference is $\Delta=r_1-r_2.$

3. Nature of loci of constant path difference

For maxima/minima, the path difference must be constant:

Bright: $\Delta=n\lambda$
Dark: $\Delta=\left(n+\frac12\right)\lambda$

Now, since $r_1-r_2=\text{constant},$ this is precisely the definition of a hyperbola in the $x$ - $z$ plane of the screen, with foci at the projections of $S_1$ and $S_2$ onto the screen.

Because the sources are symmetric about $x=0$ , the foci are symmetrically placed about $O$ in the $x$ -direction.

So the fringe pattern consists of hyperbolic bright and dark bands.

Thus, Option D is correct.

4. Approximation near the central region

Since $D\gg d$ , we may use the far-field approximation: $\Delta \approx \frac{(x+d/2)^2-(x-d/2)^2}{2\sqrt{D^2+x^2+z^2}}$ which simplifies to

\qquad R=\sqrt{D^2+x^2+z^2}.$$ Very close to $O$, $x,z\ll D$, so $$R\approx D$$ therefore $$\Delta\approx \frac{xd}{D}.$$ So near the center, constant $\Delta$ means $$x=\text{constant},$$ which are straight lines parallel to the $z$-axis (not the $x$-axis). Hence **Option C is false**. --- ## 5. Intensity at the point $O$ At $O=(0,D,0)$, $$r_1=r_2=\sqrt{D^2+\left(\frac d2\right)^2}$$ so $$\Delta=0.$$ Therefore the point $O$ is a **bright point**, not dark. Hence **Option B is false**. --- ## 6. Are the fringes semicircular? Semicircular fringes would mean curves centered at $O$ depending only on distance from $O$, i.e. only on $x^2+z^2$. But here the path difference depends asymmetrically on $x$ through $$r_1-r_2=\sqrt{\left(x+\frac d2\right)^2+D^2+z^2}-\sqrt{\left(x-\frac d2\right)^2+D^2+z^2},$$ so the loci are not circles centered at $O$. Thus **Option A is false**. --- ## 7. Final evaluation of options - **A:** False - **B:** False - **C:** False - **D:** True --- ## 8. Comparison with stored answer Stored correct answer: **A, B** My derived answer: **D** These do not match. The stored answer appears inconsistent with the interference condition $r_1-r_2=\text{constant}$ for two point sources and a screen parallel to the $XZ$-plane, which gives hyperbolic fringes and a bright point at $O$.

More from Wave Optics

1. Geometry of the setup

Let the two coherent point sources be at

S_1\left(-\frac d2,0,0\right), \qquad S_2\left(+\frac d2,0,0\right)

in the

XY

-plane.

The screen is parallel to the

XZ

-plane, so its equation is

y=D

with

D=3\,\text{m}

A general point on the screen is

P(x,D,z).

The origin

O

is the point where the line joining

S_1S_2

meets the screen, so

O=(0,D,0).

3. Nature of loci of constant path difference

For maxima/minima, the path difference must be constant:

Bright:

\Delta=n\lambda

Dark:

\Delta=\left(n+\frac12\right)\lambda

Now, since

r_1-r_2=\text{constant},

this is precisely the definition of a hyperbola in the

x

z

plane of the screen, with foci at the projections of

S_1

and

S_2

onto the screen.

Because the sources are symmetric about

x=0

, the foci are symmetrically placed about

O

in the

x

-direction.

So the fringe pattern consists of hyperbolic bright and dark bands.

Thus, Option D is correct.

4. Approximation near the central region

Since

D\gg d

, we may use the far-field approximation:

\Delta \approx \frac{(x+d/2)^2-(x-d/2)^2}{2\sqrt{D^2+x^2+z^2}}

which simplifies to

\qquad R=\sqrt{D^2+x^2+z^2}.$$ Very close to $O$, $x,z\ll D$, so $$R\approx D$$ therefore $$\Delta\approx \frac{xd}{D}.$$ So near the center, constant $\Delta$ means $$x=\text{constant},$$ which are straight lines parallel to the $z$-axis (not the $x$-axis). Hence **Option C is false**. --- ## 5. Intensity at the point $O$ At $O=(0,D,0)$, $$r_1=r_2=\sqrt{D^2+\left(\frac d2\right)^2}$$ so $$\Delta=0.$$ Therefore the point $O$ is a **bright point**, not dark. Hence **Option B is false**. --- ## 6. Are the fringes semicircular? Semicircular fringes would mean curves centered at $O$ depending only on distance from $O$, i.e. only on $x^2+z^2$. But here the path difference depends asymmetrically on $x$ through $$r_1-r_2=\sqrt{\left(x+\frac d2\right)^2+D^2+z^2}-\sqrt{\left(x-\frac d2\right)^2+D^2+z^2},$$ so the loci are not circles centered at $O$. Thus **Option A is false**. --- ## 7. Final evaluation of options - **A:** False - **B:** False - **C:** False - **D:** True --- ## 8. Comparison with stored answer Stored correct answer: **A, B** My derived answer: **D** These do not match. The stored answer appears inconsistent with the interference condition $r_1-r_2=\text{constant}$ for two point sources and a screen parallel to the $XZ$-plane, which gives hyperbolic fringes and a bright point at $O$.

Wave Optics question

Wave Optics question

1. Geometry of the setup

2. Distances from the two sources to a point on screen

3. Nature of loci of constant path difference

4. Approximation near the central region

More from Wave Optics

Wave Optics question

Wave Optics question

1. Geometry of the setup

2. Distances from the two sources to a point on screen

3. Nature of loci of constant path difference

4. Approximation near the central region

More from Wave Optics