Q.If 109 electrons move out of a body to another body every second, how much time is required to get a total charge of 1C on the other body?
Concept understanding — Photoelectric Effect
The Photoelectric Effect: When Light Knocks Electrons Loose
Imagine you're throwing tennis balls at a wall covered in loose pebbles. If you throw hard enough, a pebble might get knocked off. That's the basic picture — but the photoelectric effect is the quantum version of this, and it completely shattered classical physics.
The Intuition
Light is made of tiny packets of energy called photons. Each photon carries a specific amount of energy, determined by its colour (frequency). When a photon hits a metal surface, it can transfer its energy to an electron inside the metal. If that energy is enough, the electron breaks free and flies out.
Think of electrons in a metal like people in a room with a high window. To escape, they need enough energy to reach the window sill. A photon is like a boost — but only if it gives enough energy in one shot. No amount of weak boosts (dim light) will work if each individual boost is too small.
The Precise Statement
Ephoton=hf=ϕ+Kmax
Where:
- Ephoton=hf is the energy of a photon (Planck's constant h=6.63×10−34 J⋅s, f is frequency)
- ϕ is the work function — the minimum energy needed to remove an electron from that metal
- Kmax is the maximum kinetic energy of the ejected electron
What Classical Physics Got Wrong
Before Einstein (1905), physicists thought light was a continuous wave. They expected:
- Brighter light → more energy per electron → faster electrons
- Any colour would eventually eject electrons if you waited long enough
But experiments showed the opposite:
| Observation | Classical Prediction | Actual Result |
|---|---|---|
| Effect of intensity | Brighter light → faster electrons | Brighter light → more electrons, same speed |
| Threshold frequency | None — any light works eventually | Below a certain frequency, no electrons no matter how bright |
| Time delay | Electrons need time to absorb energy | Electrons appear instantly (within 10−9 s) |
The Key Insight
Einstein said: light behaves like a stream of particles (photons), each with energy hf. One photon interacts with one electron. If hf<ϕ, the electron cannot escape — period. If hf>ϕ, the excess energy becomes kinetic energy:
Kmax=hf−ϕ
This is why:
- Increasing intensity (more photons) ejects more electrons, but each electron still gets the same energy per photon — so their speed doesn't change.
- Below threshold frequency, even a trillion photons per second can't help — each one is too weak individually.
The photoelectric effect proved that light is quantized — it comes in discrete packets. This was the birth of quantum mechanics. Einstein won the 1921 Nobel Prize for this, not for relativity.
A Worked Example
Problem: A metal has work function ϕ=2.0 eV. Light of frequency f=6.0×1014 Hz shines on it. Find the maximum kinetic energy of ejected electrons. (h=4.14×10−15 eV⋅s)
Step 1: Photon energy
E=hf=(4.14×10−15)(6.0×1014)=2.48 eV
Step 2: Subtract work function
Kmax=2.48−2.0=0.48 eV
Step 3: Convert to joules if needed
0.48 eV×1.6×10−19=7.68×10−20 J
The electron escapes with this much kinetic energy.
Common Mistake to Avoid
Students often think "more intense light means more energy per electron." Wrong. Intensity = number of photons per second. Each photon still has the same hf. More photons = more electrons, but each electron gets the same energy kick.
The Big Picture
The photoelectric effect is your first encounter with wave-particle duality. Light, which we model as a wave for interference and diffraction, behaves as a particle when transferring energy to matter. This duality is central to all of quantum mechanics.
Final takeaway: Light ejects electrons only if each photon carries enough energy individually. The colour (frequency) determines whether ejection happens; the brightness (intensity) determines how many electrons get ejected.
"Photoelectric effect formula and Einstein equation" is among the most-searched Class 12 physics topics, and it is a core result of the Dual Nature of Radiation and Matter chapter in the NCERT/CBSE Class 12 Physics curriculum. Work function and threshold frequency questions built on this concept appear in nearly every JEE Main and NEET physics paper.
Why this formula?
Photoelectric Effect: Why the Key Formulas Hold
The photoelectric effect is a cornerstone of quantum physics. It showed that light behaves as particles (photons) , not just waves. Let's build the reasoning step-by-step.
1. The Core Idea: Energy Conservation
When a photon hits a metal surface, it transfers all its energy to a single electron inside the metal.
- The photon's energy is E=hf, where h is Planck's constant and f is the frequency of light.
- The electron needs a minimum energy to escape the metal — this is called the work function, ϕ.
Why only one electron?
Einstein proposed that light is quantized into discrete packets (photons). A single photon cannot split its energy among multiple electrons — it interacts with one electron at a time.
2. The Photoelectric Equation
If the photon's energy is greater than the work function, the excess energy becomes the electron's kinetic energy after escape:
hf=ϕ+Kmax
Where:
- hf = energy of incident photon
- ϕ = work function (minimum energy to remove electron)
- Kmax = maximum kinetic energy of ejected electron
Why "maximum" kinetic energy?
- Electrons inside the metal have different binding energies.
- Some electrons are near the surface (loosely bound) → get maximum K.
- Others are deeper → lose energy in collisions before escaping → lower K.
3. The Stopping Potential Connection
We measure Kmax using a stopping potential Vs:
Kmax=eVs
Where e is the electron charge. This is because:
- An electric field opposing the electron's motion does work eVs to stop it.
- At the stopping potential, the electron's kinetic energy is exactly balanced by the electric potential energy.
Combining:
hf=ϕ+eVs
This is the Einstein photoelectric equation in its most testable form.
4. Why the Threshold Frequency Exists
From the equation:
hf=ϕ+eVs
If f is too low, hf<ϕ. Then:
- The photon cannot supply enough energy to overcome the work function.
- No electron is ejected, regardless of light intensity.
The threshold frequency f0 is when Kmax=0:
hf0=ϕ⇒f0=hϕ
Why intensity doesn't matter for ejection?
- Intensity = number of photons per second.
- Each photon still has energy hf. If hf<ϕ, even a billion photons won't eject an electron — each photon is individually too weak.
5. Why Kinetic Energy Depends on Frequency, Not Intensity
From Kmax=hf−ϕ:
- Frequency f directly determines Kmax.
- Intensity only affects the number of electrons ejected (more photons → more electrons), not their individual energy.
This was the key experimental contradiction with classical wave theory:
- Classical: Higher intensity = bigger wave amplitude = more energy to electrons.
- Reality: Higher frequency = more energy per electron; intensity only changes current.
6. Summary of Key Relationships
| Quantity | Formula | Why it holds |
|---|---|---|
| Photon energy | E=hf | Light is quantized (Planck-Einstein) |
| Work function | ϕ=hf0 | Minimum energy to escape at threshold |
| Max kinetic energy | Kmax=hf−ϕ | Energy conservation per photon-electron |
| Stopping potential | eVs=hf−ϕ | Electric work balances kinetic energy |
| Threshold frequency | f0=ϕ/h | Below this, no ejection possible |
7. The Deeper "Why" — Particle Nature of Light
The photoelectric effect cannot be explained by classical wave theory because:
- Waves spread energy over the whole wavefront — an electron would take time to absorb enough energy.
- But experiments show instantaneous ejection (within 10−9 s).
- Wave theory predicts kinetic energy should increase with intensity — it doesn't.
Einstein's photon model resolves all three:
- Instantaneous — one photon, one interaction.
- Frequency-dependent — photon energy is hf.
- Intensity-independent — more photons = more electrons, not more energy per electron.
Key takeaway: The photoelectric effect is a direct consequence of energy quantization — both light and electron binding energy are quantized. The formulas are simply conservation laws applied to this quantum world.
Concept: Quantization of charge – charge transfer by electron motion.
Each electron carries a fundamental charge e=1.6×10−19C.
Step 1: Find the charge transferred per second.
I=n⋅e=109×1.6×10−19=1.6×10−10C/s
Step 2: To accumulate total charge Q=1C, the time required is
t=IQ=1.6×10−101=1.61010=6.25×109s
Step 3: Convert to years for perspective (optional but useful):
t=365×24×36006.25×109≈3.15×1076.25×109≈198years
The time required is 6.25×109s or approximately 198 years.
Charge accumulates at a rate determined by the number of electrons transferred per second. At 109 electrons/second, it takes approximately 200 years to accumulate 1C.
Understanding charge transfer
When electrons move from one body to another, they carry their elementary charge e=1.6×10−19C with them. The body losing electrons becomes positively charged, while the body gaining electrons becomes negatively charged. The question asks how long it takes for the receiving body to accumulate a total charge of 1C.
The key insight is that charge builds up at a constant rate if electrons transfer at a steady rate. We need to find the charge transferred per second (the current), then determine how many seconds are needed to reach 1C.
Step-by-step solution
1. Calculate the charge transferred per second
Each electron carries charge e=1.6×10−19C. If 109 electrons move every second, the charge transferred per second is:
I=n⋅e=109×1.6×10−19C/s
I=1.6×10−10C/s
This is effectively the current flowing between the two bodies.
2. Find the time required to accumulate 1C
We know that charge Q=I⋅t, where t is time. Rearranging for time:
t=IQ=1.6×10−10C/s1C
t=1.6×10−101s=1.61010s
t=6.25×109s
3. Convert to more meaningful units
This is an enormous amount of time. Let's convert to years:
t=365×24×36006.25×109years
t=3.156×1076.25×109years
t≈198years
A coulomb is a huge amount of charge at the scale of individual electrons. Even though 109 sounds like a large number, the elementary charge is so tiny that the transfer rate is extremely slow. This is why we rarely see static charges of 1C in everyday life!
Remember that 1A=1C/s. The current here is only 1.6×10−10A, which is a billionth of an ampere — far smaller than typical household currents of a few amperes.
The time required is 6.25×109s or approximately 198 years.
Instead of finding the current first, count the total number of electrons needed to make up 1C, then divide by the rate at which electrons are transferred. Same final answer: t≈6.25×109s≈198 years.
Method: Total-Electron-Count Approach
Rather than treating this as a current problem, we can think of it purely as counting: how many electrons in total does it take to reach 1C, and how long does it take to move that many electrons at the given rate?
- Find the total number of electrons needed for 1C. Each electron carries charge e=1.6×10−19C. If N electrons are needed to make up a total charge Q=1C:
N=eQ=1.6×10−191=6.25×1018electrons
-
Note the given transfer rate.
The problem states 109 electrons move across every second — this is simply electrons-per-second, a rate, independent of how we choose to compute the answer.
-
Divide total electrons by the rate to get time.
If N electrons must move at a steady rate of 109 electrons per second, the time taken is:
t=rateN=1096.25×1018s
- Simplify.
t=6.25×109s
- Sanity-check by converting to years.
t=3.15×107s/yr6.25×109≈198years
This matches the current-based method exactly, since dividing "total electrons" by "electrons per second" is algebraically the same as dividing "total charge" by "charge per second" — just counted in a different order.
This way of thinking is useful whenever a rate is given in particles per second rather than current directly — count the particles you need first, then bring in the rate.
The time required is 6.25×109s, approximately 198 years.
Common Mistakes & How to Avoid Them
Mistake 1: Using the wrong sign for charge
The error: Students often forget that electrons are negatively charged. They plug q=1.6×10−19C without considering the sign, then get confused when the time comes out negative or when they try to interpret "charge on the other body."
How to avoid:
- Always write: charge on one electron = −e=−1.6×10−19C
- When electrons leave a body, that body becomes positively charged (loss of negative charge).
- The other body receiving electrons becomes negatively charged.
- For the question: "total charge of 1C on the other body" means the other body has −1C (since it gains electrons). But the magnitude is what matters for time calculation.
Correct approach:
Use magnitude only: ∣q∣=1.6×10−19C per electron. The sign tells you the type of charge, not the amount.
Mistake 2: Confusing number of electrons with charge
The error: Students sometimes think 109 electrons per second means 109 coulombs per second, leading to absurdly short times.
How to avoid:
- Remember: Charge is quantized — every electron carries a fixed tiny charge e.
- The rate of charge transfer = (number of electrons per second) × (charge per electron)
I=(109)×(1.6×10−19)=1.6×10−10C/s
Mistake 3: Forgetting to convert units or misplacing powers of 10
The error: Students write 109×1.6×10−19=1.6×10? and mess up the exponent.
How to avoid:
- Do the exponent arithmetic carefully:
109×10−19=10−10
Then multiply by 1.6: 1.6×10−10
- Double-check: 109 is large, 10−19 is tiny — the product should be a very small number.
Mistake 4: Using the wrong formula for time
The error: Students try to use t=Q/I but plug in Q=1C and I=109 (electrons/second) directly, forgetting to convert current to coulombs/second.
How to avoid:
- Always convert to consistent units:
- Q=1C
- I=1.6×10−10C/s (after conversion)
- Then:
t=IQ=1.6×10−101=6.25×109seconds
Mistake 5: Stopping at seconds without converting to a meaningful unit
The error: Students leave the answer as 6.25×109 seconds, which is correct but not very intuitive.
How to avoid:
- Convert to years for a sense of scale:
6.25×109s÷(365×24×3600)≈198years
- This also serves as a reality check — it shows how huge 1C really is in terms of electron transfer.
Quick Summary Checklist
| Step | Common Mistake | Fix |
|---|---|---|
| Charge per electron | Wrong sign | Use magnitude 1.6×10−19C |
| Current calculation | Treat electrons as coulombs | Multiply: n×e |
| Exponent arithmetic | Wrong power of 10 | 109×10−19=10−10 |
| Time formula | Wrong I units | Convert to C/s first |
| Final answer | Leave in seconds | Convert to years for context |
Final correct answer:
t=(109electrons/s)×(1.6×10−19C/electron)1C=6.25×109s≈198years
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