Two cylindrical metallic conductors, X and Y, are connected in series across a direct-curr… — JAMB UTME

Two cylindrical metallic conductors, $X$ and $Y$ , are connected in series across a direct-current source. Wire $X$ has a diameter of $1.0\,\text{mm}$ and a free-electron density of $6.0 \times 10^{28}\,\text{m}^{-3}$ . Wire $Y$ has a diameter of $3.0\,\text{mm}$ and a free-electron density of $2.0 \times 10^{28}\,\text{m}^{-3}$ . What is the ratio of the drift velocity of free electrons in wire $X$ to that in wire $Y$ ?

$3.0$ Cevap
B
$0.33$
C
$1.0$
D
$9.0$

Cevap

The ratio of the drift velocity in wire X to that in wire Y is 3.0.

Because the wires are connected in series, the same current passes through both (

I_X = I_Y

). Using the formula for electric current in terms of drift velocity

I = n A e v_d

, where cross-sectional area

A = \pi d^2 / 4

, we find

v_d \propto 1 / (n d^2)

. Taking the ratio yields

v_X / v_Y = (n_Y d_Y^2) / (n_X d_X^2) = (2.0 \times 10^{28} \times 3.0^2) / (6.0 \times 10^{28} \times 1.0^2) = 18 / 6 = 3.0

Adım Adım Çözüm

Relate electric current to drift velocity and conductor geometry.

I = n A e v_d = n \left( \frac{\pi d^2}{4} \right) e v_d

Electric current

I

depends on free-electron density

n

, cross-sectional area

A

, elementary charge

e

, and electron drift velocity

v_d

Apply the series connection constraint.

I_X = I_Y \implies n_X d_X^2 v_X = n_Y d_Y^2 v_Y

In a series circuit, the steady current flowing through every conductor is identical.

Rearrange to solve for the drift velocity ratio

v_X / v_Y

\frac{v_X}{v_Y} = \frac{n_Y d_Y^2}{n_X d_X^2}

Isolating

v_X / v_Y

demonstrates inverse proportionality to electron density and the square of conductor diameter.

Substitute the given values into the ratio expression.

\frac{v_X}{v_Y} = \frac{(2.0 \times 10^{28}\,\text{m}^{-3}) \times (3.0\,\text{mm})^2}{(6.0 \times 10^{28}\,\text{m}^{-3}) \times (1.0\,\text{mm})^2} = \frac{2.0 \times 9.0}{6.0 \times 1.0} = \frac{18.0}{6.0} = 3.0

Numerical calculation yields the simplified dimensionless ratio.

Anahtar Kavram

Drift Velocity and Current Density in Series Conductors

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