When a current flows in an aluminum wire of diameter 2.91 mm 2.91 mm , the drift speed of the conduction electrons is 0.000191 m/s 0.000191 m/s . How many electrons are flowing past a given point each second? The conduction electron density in aluminum is 6.00 × 10 28 m − 3 6.00×1028 m−3 .

Answer :

Answer:

Number of electrons are flowing per second is 2.42 x 10¹⁹

Explanation:

The electric current flows through a wire is given by the relation :

[tex]I=envA[/tex]   ....(1)

Here I is current, e is electronic charge, v is drift velocity of electrons and A is the Area of the wire.

But electric current is also define as rate of electrons passing through junction times their charge, i.e. ,

[tex]I=Ne[/tex]      ....(2)

Here N is the rate of electrons passing through junction.

From equation (1) and (2).

[tex]eN = envA[/tex]

[tex]N=nvA[/tex]

But area of wire, [tex]A=\pi \frac{d^{2} }{4}[/tex]

Here d is diameter of wire.

So, [tex]N = nv\pi \frac{d^{2} }{4}[/tex]

Substitute 2.91 x 10⁻³ m for d, 0.000191 m/s for v and 6 x 10²⁸ m⁻³ for n in the above equation.

[tex]N = 6\times10^{28}\times 0.000191\times\pi \frac{(2.91\times10^{-3} )^{2} }{4}[/tex]

N = 2.42 x 10¹⁹ s⁻¹  

There are [tex]2.42\times10^{19}[/tex] electrons flowing past a given point each second.

Conduction current:

The current I in a wire with cross-sectional area A is given by:

[tex]I=nev_dA[/tex]

where n is the conduction electron density = 6×10²⁸m⁻³

[tex]v_d[/tex] is the drift velocity = 0.000191 m/s

A is the area = [tex]\frac{\pi d^2}{4}[/tex] ( here d is the diameter = 0.000291m)

In terms of rate of flow of charge the current can be expressed as:

[tex]I=Ne[/tex]

where N is the number of electrons passing through a point per unit time.

Equating both the expressions we get:

[tex]N=nv_dA\\\\N=nv_d(\frac{\pi d^2}{4})\\\\N=6\times10^{28}\times0.000191\times(\frac{3.14\times(0.00291)^2}{4})\\\\N=2.42\times10^{19}\;per\;second[/tex]

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