Irodov 3.207: The Momentum Hidden in a Current – Amit & Deepali Sinha Foundation

Problem Irodov 3.207

A straight wire of length $l = 1000$ m carries a current $I = 70$ A. Find the total momentum of the electrons in the wire.

The straight wire of length l carrying the current I. — The straight wire of length $l$ carrying the current $I$ .

Given

$l = 1000$ m
$I = 70$ A

Find

the total momentum

p

of the electrons

Solution

Step 1The current in terms of the drift

Let the wire have cross-section $S$ and $n$ free electrons per unit volume, drifting with the mean velocity $v$ . In one second the electrons in a length $v$ of the wire pass a cross-section, so the current is

I = n e v S \tag{1}

where $e$ is the magnitude of the electron’s charge.

Step 2Adding up the momenta

The wire holds $nSl$ free electrons, each carrying the drift momentum $mv$ along the wire, with $m$ the electron mass. Their total momentum, regrouped to bring in (1):

p = nSl\cdot mv = \frac{m\,l}{e}\,(nevS)

Neither $n$ nor $S$ survives on its own: only the current is needed. Numerically, with $m = 9.11\times10^{-31}$ kg and $e = 1.60\times10^{-19}$ C:

p = \frac{(9.11\times10^{-31}~\text{kg})(1000~\text{m})(70~\text{A})}{1.60\times10^{-19}~\text{C}}

p = \frac{m\,l\,I}{e} \approx 0.40~\mu\text{N}\cdot\text{s}

(2)

It points along the wire, opposite to the current, because the electrons carry negative charge.

Check

Dimensions: $m\,l\,I/e$ is kg $\cdot$ m $\cdot$ A/C, and A/C is 1/s, so it is kg $\cdot$ m/s = N $\cdot$ s. Mass flux view: $mI/e$ is the mass of electrons crossing a section each second, which is the electron mass per unit length times their speed, that is their momentum per unit length; times $l$ it is the total, whatever $n$ , $v$ and $S$ are separately. Reversing $I$ reverses $p$ , as it must.

Remark

The result depends only on the product $Il$ , the same current element that fixes the force on the wire in a magnetic field: the factor $m/e$ converts it into momentum. It is tiny, under a micronewton second for a kilometre of heavy current, which is why the inertia of the electrons shows up only in delicate experiments such as stopping a spinning coil.