Physics and Mathematics
Quantization of Electric Charge
1. Statement of the Concept
Electric charge is quantized, meaning it always exists in discrete, indivisible packets.
The smallest unit of charge is the elementary charge:
[ e = 1.6 \times 10^{-19} \text{ C} ]
Therefore, any observable charge must be an integer multiple of [e]:
[ q = \pm n e \quad \text{(where n is an integer)} ]
2. Clear Explanation and Mathematical Derivation
Meaning of Quantization
Quantization means charge is not continuous — it cannot take any arbitrary value.
Atoms are composed of:
- Electrons → charge [ -e ]
- Protons → charge [ +e ]
Thus, any object can only have a total charge equal to an integer multiple of [e].
Derivation from Atomic Structure
If an object gains or loses electrons, the total charge change is:
[ q = \Delta N \cdot e ]
Where:
- [\Delta N] = number of electrons gained or lost
- [e] = elementary charge
Since [\Delta N] is always an integer, charge can only change in steps of [e].
Why Is Charge Quantized?
- Because electrons are the smallest carriers of free charge.
- Fractional electrons have never been observed.
- Even if quarks have fractional charges, they are never found isolated → only in combinations giving multiples of [e].
Macroscopic Continuity Illusion
Objects have extremely large numbers of electrons ((~10^{19}) or more).
So charge increments of [e] are too tiny to notice, creating an illusion of continuous charge.
3. Dimensions and Units
| Quantity | Unit | Symbol | Dimension |
|---|---|---|---|
| Charge | Coulomb | C | [IT] |
4. Key Features of Quantization
- Smallest free charge = [e]
- All charges are multiples of [e]
- No fractional charge is ever observed in isolation
- Origin lies in the atomic structure of matter
- Large macroscopic charges appear continuous
- Charge transfer always occurs in integer steps (electrons moving)
5. Important Formulas to Remember
| Concept | Formula |
|---|---|
| Quantization of charge | [ q = \pm n e ] |
| Elementary charge | [ e = 1.6 \times 10^{-19} \text{ C} ] |
| Charge due to electron transfer | [ q = \Delta N e ] |
| Number of electrons transferred | [ \Delta N = \dfrac{q}{e} ] |
6. Conceptual Questions with Solutions
1. Why does charge always appear in multiples of [e]?
Because electrons carry the smallest possible charge magnitude [e]. Any transfer of charge involves gaining or losing whole electrons, not fractions.
2. Why do we not observe fractional charges like [0.5e]?
Because electrons and protons have fixed charges. Fractional charges exist only in quarks, but quarks never appear alone in nature.
3. How can macroscopic charge appear continuous if it is quantized?
Because the value of [e] is extremely small, and charged bodies contain huge numbers of electrons. Quantization steps become unnoticeable.
4. Can charge ever be [2.5e]?
No. The number of electrons transferred must be an integer, so charge must be an integer multiple of [e].
5. If you transfer 500 electrons to an object, what kind of charge does it get?
It becomes negatively charged because electrons were added.
6. Why is it impossible to create charge less than [1.6×10⁻¹⁹ C]?
Because electrons cannot be subdivided; they are elementary particles.
7. Can a body have zero charge even though it contains charged particles?
Yes. When positive and negative charges are equal, net charge is zero.
8. Does quantization apply to both static and moving charge?
Yes. Whether at rest or in motion (electric current), charge is always quantized.
9. Why does removal of an electron give +e but not +0.5e?
Because electrons carry an indivisible charge of exactly [e].
10. Does charge become quantized even in conductors?
Yes. Even though free electrons move easily in conductors, each electron still carries a fixed charge [e].
7. FAQ / Common Misconceptions
1. “Quantization means charge is always small.”
No. Charge can be extremely large, but still in multiples of [e].
2. “Fractional charges do not exist.”
They exist only in quarks, but quarks cannot be isolated, so fractional charge is never observable.
3. “Quantization means charge is discontinuous even in large objects.”
On atomic scale yes, but macroscopically charge appears continuous because [e] is extremely small.
4. “You can transfer half an electron’s charge.”
Impossible. Electrons are fundamental and indivisible.
5. “Neutral objects contain no charges.”
Incorrect. They contain equal positive and negative charges.
6. “Quantization applies only to solids.”
It applies to all matter because electrons exist in all atoms.
7. “Charge can be continuously created.”
No. Charge is transferred, not created; and always in integer multiples of [e].
8. “Loss of 0.2e is possible.”
No. You cannot lose a fraction of an electron.
9. “Charge can exist without matter.”
False. Charge is a property of matter.
10. “Current is continuous, so charge must be continuous.”
Current is continuous because enormous numbers of discrete electrons flow per second.
8. Practice Questions (with Step-by-Step Solutions)
1. How many electrons constitute a charge of [3.2 \times 10^{-19} \text{ C}]?
Step 1: Use
[ n = \dfrac{q}{e} ]
Step 2:
[ n = \dfrac{3.2 \times 10^{-19}}{1.6 \times 10^{-19}}] [= 2 ]
Answer: 2 electrons
2. A body has a charge of [ -4.8 \times 10^{-19} \text{ C} ]. How many electrons does it have in excess?
[ n = \dfrac{4.8 \times 10^{-19}}{1.6 \times 10^{-19}} = 3 ]
Answer: 3 extra electrons
3. A neutral body gains [5 \times 10^{10}] electrons. What is the resulting charge?
Step 1:
[ q = -ne ]
Step 2:
[ q = -5 \times 10^{10} \times 1.6 \times 10^{-19} ]
Step 3:
[ q = -8 \times 10^{-9} \text{ C} ]
Answer: [ -8 \times 10^{-9} \text{ C} ]
4. Can a body have a charge of [5.5 \times 10^{-19} \text{ C}]? Explain.
Step 1: Check if divisible by [e]:
[ n = \dfrac{5.5 \times 10^{-19}}{1.6 \times 10^{-19}}] [= 3.4375 ]
Not an integer → impossible
Answer: No, charge must be integer multiple of [e].
5. A body loses [10^{12}] electrons. What is its new charge?
Step 1: Loss of electrons → positive charge
[ q = +ne ]
Step 2:
[ q = 10^{12} \times 1.6 \times 10^{-19} ]
Step 3:
[ q = 1.6 \times 10^{-7} \text{ C} ]
Answer: [ +1.6 \times 10^{-7} \text{ C} ]
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