Q.(a) Explain the meaning of the statement 'electric charge of a body is quantised'.

1. The statement means that electric charge on any body is always an integer multiple of the elementary charge $e = 1.6 \times 10^{-19} \, \text{C}$. Charge cannot exist in arbitrary fractions of $e$ — it comes in discrete packets. If a body has $n$ excess or deficit electrons, its net charge is $q = \pm n e$, where $n$ is an integer.
2. At the macroscopic scale, the total charge involved is typically of the order of microcoulombs ($\mu\text{C}$) or more. For example, $1 \, \mu\text{C}$ corresponds to about $6.25 \times 10^{12}$ electrons. …

(a) What "quantised" means for electric charge

The statement "electric charge of a body is quantised" means that any observable charge $Q$ on an object is always an integer multiple of a fundamental, smallest unit of charge. That unit is the magnitude of the charge on a single electron or proton, denoted by $e$:

So if a body has a net charge $Q$, it must satisfy:

You cannot have, say, $0.5e$ or $1.7e$ on a body in isolation. Charge comes in discrete packets — it is not a continuous fluid that can be divided arbitrarily. This was first demonstrated convincingly by Millikan's oil drop experiment, which showed that every measured charge was a multiple of $e$.

(b) Why we ignore quantisation at macroscopic scales

When dealing with macroscopic (large-scale) charges — say, a charged metal sphere carrying $1 \, \mu\text{C}$ — the number of excess electrons (or protons) is enormous. Let's calculate:

That's over six trillion elementary charges. Now, if you add or remove even a few billion electrons, the change in charge is:

This is $0.16$ nanocoloumbs — far below the sensitivity of most macroscopic measuring instruments. The relative jump between allowed charge values is:

$\frac{e}{Q} \approx \frac{1.6 \times 10^{-19}}{10^{-6}} = 1.6 \times 10^{-13}$ …

Method: Quantisation of Charge — Explanation & Justification

This is a concept-based reasoning method, not a calculation. The steps below apply to both parts (a) and (b).

(a) Meaning of “electric charge of a body is quantised”

Step 1 – State the fundamental idea

Electric charge cannot exist in arbitrary amounts. It always comes in discrete packets.

Step 2 – Identify the smallest unit

The smallest possible free charge is the magnitude of the electron’s charge:

Step 3 – Express the quantisation condition

Any observable charge $Q$ on a body must be an integer multiple of $e$:

Step 4 – Give the physical reason

This arises because charge is carried by electrons and protons — indivisible particles (under normal conditions). You cannot have half an electron.

Key result: Charge is quantised means $Q = n e$, with $n$ an integer.

(b) Why quantisation is ignored for macroscopic charges

Step 1 – Compare magnitudes

Macroscopic charges (e.g., on a comb, a capacitor plate) are typically of the order of microcoulombs ($\mu$C) or more.

Step 2 – Find the number of electrons involved

Number of electrons in $1 \, \mu\text{C}$: …

Here are the common mistakes students make on this question about quantisation of charge in the Photoelectric Effect chapter, along with how to avoid each.

Mistake 1: Confusing "Quantisation" with "Conservation"

• The Error: Students often write that "charge is quantised" means "charge cannot be created or destroyed." This is the law of conservation of charge, not quantisation.
• Why it happens: Both concepts appear in the same chapter (Photoelectric Effect / Dual Nature), and the words sound similar.
• How to avoid: Remember the keyword: "packets" or "multiples."
  • Quantisation = charge exists only in discrete packets (multiples of $e$).
  • Conservation = total charge in an isolated system remains constant.

Mistake 2: Forgetting the Exact Value of the Fundamental Charge

• The Error: Stating that charge is quantised in multiples of "the charge of an electron" but not giving the numerical value, or writing the wrong value (e.g., $1.6 \times 10^{-19} \text{ C}$ is correct; $1.6 \times 10^{-19} \text{ J}$ is wrong).
• Why it happens: Rushing through the definition without memorising the constant.
• How to avoid: Always write the exact statement:

  "Electric charge exists only in discrete packets which are integral multiples of the elementary charge $e = 1.6 \times 10^{-19} \text{ C}$."

  • Formula to memorise: $q = \pm n e$, where $n = 0, 1, 2, 3, \dots$

Mistake 3: Writing "Charge is Quantised" Without the Integral Multiple Condition

• The Error: Saying "charge is quantised" but not specifying that it must be an integer multiple of $e$. Some students write "any multiple" (implying fractional multiples like $0.5e$ are allowed).
• Why it happens: Overlooking the word "integral" in the NCERT definition.
• How to avoid: Explicitly state: $q = \pm n e$, where $n$ is an integer ($n = 0, 1, 2, \dots$). No fractional $n$ is allowed.

Mistake 4: Forgetting the "Why" for Macroscopic Charges (Part b)

• The Error: Simply saying "because charges are large" without explaining why we can ignore quantisation.
• Why it happens: Students memorise the answer but don't understand the logic.
• How to avoid: Use the analogy of grains of sand:
  • A single grain of sand is like $e$ (tiny).
  • A bucket of sand is like a macroscopic charge (huge).
  • When you measure the bucket's mass, you don't count individual grains — the discreteness is negligible compared to the total.
  • Key line: "Since $e$ is very small ($1.6 \times 10^{-19} \text{ C}$), a macroscopic charge contains an enormous number of electrons ($n \approx 10^{19}$ or more). The addition or removal of a few electrons causes an imperceptible change, so the charge appears continuous."

Mistake 5: Mixing Up "Quantisation of Charge" with "Quantisation of Energy" …