Interference — constructive, destructive conditions: Question bank

1.6.17 · D5 · Physics › Oscillations & Waves › Interference — constructive, destructive conditions

Definition Is page pe use kiye gaye symbols

Traps se pehle, har symbol ko clearly samajh lo taaki koi line undefined letter use na kare.

$Δ x$ = path difference: wo extra distance jo ek wave dusri ke muqable mein travel karti hai milne se pehle — metres mein measure hoti hai.
$ϕ$ = phase difference: do waves kitni out of step hain, radians (ya degrees) mein measure hota hai. Path se linked hai $ϕ = \frac{2 π}{λ} Δ x$ ke zariye, isliye ek poora wavelength ka extra path ( $Δ x = λ$ ) ek full cycle ( $ϕ = 2 π$ ) hota hai.
$λ$ = wavelength: ek full wave cycle ki length.
$a_{1}, a_{2}$ = do individual waves ke amplitudes (peak displacements). Jab ye equal hon to hum sirf $a$ likhte hain.
$A$ = do waves ke add hone ke baad resultant amplitude.
$I_{0}$ = akele ek wave ki intensity (brightness/loudness), jahan $I_{0} \propto a^{2}$ .
$I$ = resultant intensity, jahan $I \propto A^{2}$ .

True or false — justify

Two waves always reinforce when their crests meet

True — crest ka crest se milna matlab displacements same sign ke saath add hote hain, isliye resultant bada hota hai. Yahi exactly constructive case hai, path difference pura wavelength ka whole number.

Interference requires two waves of the same amplitude

False — unequal amplitudes phir bhi interfere karte hain (parent mein Example 3), lekin kabhi bhi zero tak cancel nahi ho sakte; minimum resultant amplitude

∣ a_{1} - a_{2} ∣

hota hai (ek nonzero displacement),

0

nahi.

Two independent light bulbs can produce a stable interference pattern

False — independent sources randomly phase drift karte hain, isliye phase difference constant nahi rehta; pattern wash out ho jaata hai. Stable fringes ke liye coherent sources chahiye. Dekho Coherence and Path Difference.

At a destructive point the energy of the two waves is destroyed

False — energy sirf redistribute hoti hai; bright points pe incoherent sum

2 I_{0}

se excess (extra

2 I_{0}

, kyunki

4 I_{0} - 2 I_{0} = 2 I_{0}

) dark points pe

2 I_{0}

ki kami ko exactly compensate karta hai. Spatial average

2 I_{0}

rehta hai. Dekho Energy in Waves.

If you add the two intensities you always get the true intensity

False — intensities sirf incoherent sources ke liye add hoti hain. Coherent waves ke liye pehle amplitudes add karo, phir square karo, jo cross term

2 a_{1} a_{2} cos ϕ

deta hai.

A phase difference of

2 π

gives the same result as a phase difference of

0

True — phase

2 π

period ke saath cyclic hota hai, isliye

ϕ = 2 π

aur

ϕ = 0

dono ka matlab "perfectly in step" hai aur dono maximum intensity dete hain.

Constructive interference violates conservation of energy because

4 I_{0} > I_{0} + I_{0}

False — ye sirf ek bright point ki intensity hai; poore pattern pe average karo to intensity

2 I_{0}

hai, exactly do sources ka sum.

Zero path difference gives a dark fringe

False — zero path difference matlab perfectly in step, sabse bright central maximum (

n = 0

Δ x = 0

). Ye classic "

λ

is the condition" trap hai.

Interference and superposition are the same thing

Mostly false as stated — superposition woh underlying addition rule hai jo har instant pe apply hoti hai; interference woh rule hai jo fixed pattern of maxima aur minima ke roop mein observe hota hai. Interference Principle of Superposition ka ek special, observable consequence hai.

Spot the error

"Destructive interference needs

Δ x = λ

Wrong —

Δ x = λ

constructive hai (whole wavelength = back in step). Destructive ke liye half-integer extra chahiye:

Δ x = (n + \frac{1}{2}) λ

jahan

n = 0, \pm 1, \pm 2, \dots

"Since

I = 4 I_{0} cos^{2} (ϕ /2)

, the maximum intensity is

2 I_{0}

Wrong —

cos^{2}

maximum

1

pe hota hai, jo

I_{m a x} = 4 I_{0}

deta hai.

2 I_{0}

average hai, jo

ϕ = π /2

(yaani

9 0^{\circ}

) pe milta hai, maximum nahi.

"Two waves with amplitudes

3 a

and

4 a

in antiphase cancel to zero."

Wrong — full cancellation ke liye equal amplitudes chahiye. Yahan minimum resultant amplitude

∣4 a - 3 a ∣ = a

hai (isliye minimum intensity

\propto a^{2}

hai, zero nahi).

"For interference we add the frequencies of the two waves."

Wrong — hum displacements add karte hain, aur stable pattern ke liye do waves ki same frequency honi chahiye. Alag frequencies add karne par Beats milte hain, ek time pattern, fixed spatial pattern nahi.

ϕ = (2 π / λ) Δ x

, so a large wavelength gives a large phase difference."

Wrong direction —

λ

denominator mein hai, isliye bada

λ

same path ke liye chhota phase deta hai. Ek wavelength ka path hamesha

2 π

of phase ke barabar hota hai, chahe

λ

kuch bhi ho.

"At

ϕ = 9 0^{\circ}

the waves fully cancel."

Wrong —

cos^{2} (4 5^{\circ}) = \frac{1}{2}

, isliye

I = 2 I_{0}

, neutral average. Full cancellation ke liye

ϕ = 18 0^{\circ}

(yaani

π

radians) chahiye.

Why questions

Why do we add amplitudes and not intensities for coherent waves?

Kyunki woh physical quantity jo superpose hoti hai woh displacement (amplitude) hai, instant by instant; intensity (

\propto A^{2}

) ek derived, squared quantity hai aur squaring sum pe distribute nahi hoti.

Why does one wavelength of path difference correspond to exactly

2 π

of phase?

Ek full wavelength wave ka ek complete cycle hai, aur ek complete cycle

2 π

radians of phase hai — master link

ϕ = (2 π / λ) Δ x

sirf path ko us cycle count mein scale karta hai.

Why is intensity proportional to amplitude squared and not amplitude itself?

Wave ke dwara carry ki gayi energy displacement ke square pe depend karti hai (jaise ek swinging particle mein kinetic energy

\propto v^{2}

), isliye brightness/loudness

A^{2}

ke scale karta hai. Dekho Energy in Waves.

Why does the cross term

2 a_{1} a_{2} cos ϕ

carry all the "interference"?

a_{1}^{2}

aur

a_{2}^{2}

pieces sirf do waves ki apni intensities hain; sirf

cos ϕ

term unke relative phase pe depend karta hai, positive (reinforce) ya negative (cancel) swing karta hai. Coherence khatam karo aur

cos ϕ

zero pe average ho jaata hai, plain

I_{0} + I_{0}

bacha rehta hai.

Why must sources be coherent for a visible pattern?

Aankhein/detector time ke saath average karte hain; agar

ϕ

randomly jitter kare, to

cos^{2} (ϕ /2)

term har jagah

\frac{1}{2}

pe average ho jaata hai, jo bina bright aur dark fringes ke uniform

2 I_{0}

deta hai.

Why does the average intensity across the whole pattern equal

2 I_{0}

Kyunki energy conserved hai: bright points pe excess (

4 I_{0}

) dark points pe deficit (

0

) ko exactly balance karta hai, aur

cos^{2}

ka mean ek full cycle pe

\frac{1}{2}

hota hai, jo

4 I_{0} \times \frac{1}{2} = 2 I_{0}

deta hai.

Why can a standing wave be seen as interference?

Opposite directions mein travel karti do identical waves superpose hoti hain; unka fixed pattern of nodes (hamesha destructive) aur antinodes (hamesha constructive) space mein frozen interference hai. Dekho Standing Waves.

Edge cases

What is the intensity when the two waves have a phase difference of exactly

π /2

(that is

9 0^{\circ}

I = 4 I_{0} cos^{2} (π /4) = 2 I_{0}

— neutral "average" point, na maximum na minimum.

What happens to interference if one source is switched off?

Superpose karne ke liye koi doosri wave nahi, isliye koi pattern nahi banta — aapko har jagah single source ki plain, uniform intensity

I_{0}

milti hai.

What is the resultant when two equal waves are exactly

18 0^{\circ}

(that is

π

radians) out of phase?

Ye completely cancel ho jaate hain:

A = 2 a cos (9 0^{\circ}) = 0

, isliye

I = 0

. Ye perfect destructive interference hai, sirf equal amplitudes ke saath possible hai.

At the central point of Young's Double Slit Experiment, what is the path difference and what fringe appears?

Path difference zero hai (dono slits se equal distances), isliye ye central bright maximum hai,

n = 0

If two coherent waves have a very small amplitude difference, can you ever get total darkness?

Nahi — minimum resultant amplitude

∣ a_{1} - a_{2} ∣

hai, jo nonzero hai chahe difference kitna bhi chhota ho. Sirf exactly equal amplitudes true zero dete hain.

What does the interference pattern look like for two waves of slightly different frequency?

Koi fixed spatial pattern nahi; balki resultant amplitude time mein rise aur fall karta hai — ye Beats hain, interference ke temporal cousin.

Recall check

Recall Which quantity superposes — amplitude or intensity?

Amplitude (displacement). Intensity baad mein resultant amplitude ko square karke derive hoti hai.

Recall Fastest way to classify a path difference

$Δ x$ ko $λ$ se compare karo: agar $Δ x = nλ$ (wavelengths ka whole number) → constructive (Wow); agar $Δ x = (n + \frac{1}{2}) λ$ (half-integer, $n = 0, \pm 1, \pm 2, \dots$ ) → destructive (Hush).

Connections

Interference — constructive, destructive conditions — woh parent jise ye bank drill karta hai.
Principle of Superposition — woh add-displacements rule jo har trap test karta hai.
Coherence and Path Difference — kyon independent sources fail hote hain.
Young's Double Slit Experiment — jahan central-fringe edge case rehta hai.
Standing Waves — counter-propagating waves ka interference.
Beats — time mein different frequencies ka interference.
Energy in Waves — kyon $I \propto A^{2}$ aur energy conserved hai.