Is the momentum conserved when charge crosses a junction in an electric circuit? Why or why not?

NCERT Exemplar · Q9

Q. Is the momentum conserved when charge crosses a junction in an electric circuit? Why or why not?

Uttarakhand UbseSubjective· 2mImportance★★★★★

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Concept understanding — Drift Velocity

Drift Velocity: The Slow March of Electrons

Electrons in a metal are always moving — but randomly. At room temperature they zip around at roughly $10^6$ m/s, colliding with the lattice ions every few trillionths of a second. Without an electric field this motion cancels out: for every electron heading left another heads right, so the net velocity is zero.

Apply a battery and the field gives every electron a tiny, steady push in one direction. Between collisions the electron accelerates only briefly before smashing into an ion and losing its directed motion. What survives is a very small average velocity along the field — the drift velocity.

Note

The random thermal speed is about $10^5$ m/s, but the drift velocity is only about $10^{-4}$ m/s — about a billion times slower. An electron drifts slower than a snail, yet a lamp lights instantly, because the electric field (not the electrons) propagates at nearly the speed of light and starts every electron drifting almost at once.

The precise definition

Drift velocity ( $v_d$ ) is the average velocity acquired by the charge carriers in a conductor under an applied electric field:

v_d = \frac{eE\tau}{m}

where:

$e$ = electron charge ( $1.6 \times 10^{-19}$ C)
$E$ = electric field inside the conductor (V/m)
$\tau$ = average relaxation time — the mean time between collisions (s)
$m$ = electron mass ( $9.1 \times 10^{-31}$ kg)

v_d = \frac{eE\tau}{m}

Linking to current

Drift velocity connects the microscopic motion of electrons to the current an ammeter reads:

I = n e A v_d

where $n$ is the free-electron number density and $A$ the cross-sectional area. A larger $v_d$ means more current, but $v_d$ stays tiny because $\tau$ is tiny (about $10^{-14}$ s in copper).

Tip

For a copper wire carrying $1$ A with area $1$ mm² and $n \approx 8.5 \times 10^{28}$ m $^{-3}$ :

$v_d = \frac{I}{neA} \approx \frac{1}{(8.5\times10^{28})(1.6\times10^{-19})(10^{-6})} \approx 7 \times 10^{-5}\ \text{m/s}$ …

Why this formula?

Drift Velocity: Why the Formula Holds

Let's build this from first principles — understanding why electrons drift the way they do, not just memorizing the formula.

1. The Core Idea: What is Drift Velocity?

In a conductor, free electrons are constantly moving randomly (thermal motion, speeds ~ $10^5$ m/s). Without an electric field, their net displacement is zero — they're like a swarm of bees buzzing in all directions.

When we apply an electric field $\vec{E}$ , it gently nudges each electron in the opposite direction (since electrons are negatively charged). This small, steady net velocity superimposed on the random motion is drift velocity ( $v_d$ ).

Key insight: Drift velocity is not the speed of individual electrons — it's the average velocity of the entire electron cloud.

2. The Derivation: Step by Step

Step 1: Force on a single electron

An electron of charge $-e$ in an electric field $\vec{E}$ experiences:

\vec{F} = -e \vec{E}

The magnitude of acceleration (opposite to $\vec{E}$ ) is:

a = \frac{F}{m} = \frac{eE}{m}

where $m$ is the electron's mass.

Step 2: What happens between collisions?

Electrons don't accelerate forever — they keep colliding with atoms/ions in the metal lattice. Let the average time between collisions be $\tau$ (relaxation time). Just after a collision an electron's velocity is essentially random (zero average in the field direction); it then accelerates for time $\tau$ before the next collision.

Step 3: Drift velocity

Averaging the field-driven velocity over the relaxation time $\tau$ gives the net drift:

\boxed{v_d = \frac{eE\tau}{m}}

Here $\tau$ is the average time since the last collision, so this expression already averages over electrons at every stage between collisions — it is the standard result used in the NCERT treatment.

3. Connecting to Current: The Big Picture

Drift velocity directly gives us current density $J$ :

J = n e v_d

where $n$ = number of free electrons per unit volume. …

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Momentum is not conserved when charge crosses a junction because the charge carriers (electrons) experience a net force from the electric field, and the junction itself exerts an impulse on them — the system is not isolated.

Why this question matters

This is a subtle question that tests whether you understand the difference between charge conservation (which always holds at a junction — Kirchhoff’s Current Law) and momentum conservation (which requires a closed, isolated system). Many students confuse the two because both involve “flow” and “continuity.” Let’s separate them clearly.

The core idea: Drift velocity and the electric field

Inside a conductor, electrons move with a drift velocity $\vec{v}_d$ — a slow, net motion superimposed on their random thermal jitter. This drift is caused by the electric field $\vec{E}$ inside the wire. The field exerts a force $\vec{F} = -e \vec{E}$ on each electron.

At a junction, the wire geometry changes. The current $I$ (rate of flow of charge) is the same in all branches — that’s charge conservation. But the drift velocity depends on the cross-sectional area $A$ and the number density $n$ of free electrons:

I = n e A v_d

So if the wire splits into two thinner wires, $A$ decreases, and $v_d$ must increase to keep $I$ constant. That means the electrons speed up or slow down as they pass through the junction.

Step-by-step reasoning

Momentum of a single charge carrier

An electron of mass $m_e$ moving with drift velocity $v_d$ has momentum $p = m_e v_d$ . At a junction, if the wire cross-section changes, $v_d$ changes, so the electron’s momentum changes.
Net force on the electron

A change in momentum requires a net force ( $\vec{F} = d\vec{p}/dt$ ). That force comes from the electric field inside the conductor. But the electric field is not the only player — the lattice ions in the wire also exert forces on the electrons through collisions. When an electron collides with an ion, momentum is transferred to the lattice.
The system is not isolated

For momentum to be conserved, the system must have no external net force. Here, the “system” of moving charges is not isolated:
- The battery maintains an electric field (external force).
- The wire’s lattice exerts forces during collisions.
- The junction itself is a physical constraint that redirects charges.
What about the whole circuit?

If you take the entire circuit (battery + wires + resistors) as your system, then momentum is conserved overall — but only if you include the lattice and the battery. The charges alone do not conserve momentum.
A concrete example

Suppose a current $I$ flows in a thick wire of area $A_1$ , then splits into two identical thin wires each of area $A_2 = A_1/2$ .
- Before the junction: $v_{d1} = I/(n e A_1)$
- After the junction (each branch): $v_{d2} = I/(n e A_2) = 2 v_{d1}$ Each electron doubles its speed, so its momentum doubles. That extra momentum came from the electric field and the lattice forces at the junction.

Watch out

Do not confuse charge conservation with momentum conservation.

Charge is conserved at a junction: $\sum I_{\text{in}} = \sum I_{\text{out}}$ (Kirchhoff’s Current Law). …

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Q.Is the momentum conserved when charge crosses a junction in an electric circuit? Why or why not?

Concept understanding — Drift Velocity

Drift Velocity: The Slow March of Electrons

The precise definition

Linking to current

Why this formula?

Drift Velocity: Why the Formula Holds

1. The Core Idea: What is Drift Velocity?

2. The Derivation: Step by Step

Step 1: Force on a single electron

Step 2: What happens between collisions?

Step 3: Drift velocity

3. Connecting to Current: The Big Picture

Why this question matters

The core idea: Drift velocity and the electric field

Step-by-step reasoning

Method: Reasoning About Conservation Laws in a Circuit

Steps

Q.Is the momentum conserved when charge crosses a junction in an electric circuit? Why or why not?

Why this question matters

The core idea: Drift velocity and the electric field

Step-by-step reasoning

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