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A current of 4.00 mA flows through a copper wire. The wire has an initial diameter of 4.00 mm which gradually tapers to a diameter of 1.00 mm. The wire length is
2.00 m and copper has a number density of 8.50 × 1028 m–3.

Find the change in mean drift velocity for electrons as they pass from one end of the wire to the other and therefore calculate the average acceleration of the electrons.

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The change in mean drift velocity for electrons as they pass from one end of the wire to the other is 3.506 x 10⁻⁷ m/s and average acceleration of the electrons is 4.38 x 10⁻¹⁵ m/s².

The given parameters;

  • Current flowing in the wire, I = 4.00 mA
  • Initial diameter of the wire, d₁ = 4 mm = 0.004 m
  • Final diameter of the wire, d₂ = 1 mm = 0.001 m
  • Length of wire, L = 2.00 m
  • Density of electron in the copper, n = 8.5 x 10²⁸ /m³

The initial area of the copper wire;

[tex]A_1 = \frac{\pi d^2}{4} = \frac{\pi \times (0.004)^2}{4} =1.257\times 10^{-5} \ m^2[/tex]

The final area of the copper wire;

[tex]A_2 = \frac{\pi d^2}{4} = \frac{\pi (0.001)^2}{4} = 7.86\times 10^{-7} \ m^2[/tex]

The initial drift velocity of the electrons is calculated as;

[tex]v_d_1 = \frac{I}{nqA_1} \\\\v_d_1 = \frac{4\times 10^{-3} }{8.5\times 10^{28} \times 1.6\times 10^{-19} \times 1.257\times 10^{-5}} \\\\v_d_1 = 2.34 \times 10^{-8} \ m/s[/tex]

The final drift velocity of the electrons is calculated as;

[tex]v_d_2 = \frac{I}{nqA_2} \\\\v_d_2 = \frac{4\times 10^{-3} }{8.5\times 10^{28} \times 1.6\times 10^{-19} \times 7.86\times 10^{-7}} \\\\v_d_2 = 3.74\times 10^{-7} \ m/s[/tex]

The change in the mean drift velocity is calculated as;

[tex]\Delta v = v_d_2 -v_d_1\\\\\Delta v = 3.74\times 10^{-7} \ m/s \ -\ 2.34 \times 10^{-8} \ m/s = 3.506\times 10^{-7} \ m/s[/tex]

The time of motion of electrons for the initial wire diameter is calculated as;

[tex]t_1 = \frac{L}{v_d_1} \\\\t_1 = \frac{2}{2.34\times 10^{-8}} \\\\t_1 = 8.547\times 10^{7} \ s[/tex]

The time of motion of electrons for the final wire diameter is calculated as;

[tex]t_2 = \frac{L}{v_d_1} \\\\t_2= \frac{2}{3.74 \times 10^{-7}} \\\\t_2 = 5.348 \times 10^{6} \ s[/tex]

The average acceleration of the electrons is calculated as;

[tex]a = \frac{\Delta v}{\Delta t} \\\\a = \frac{3.506 \times 10^{-7} }{(8.547\times 10^7)- (5.348\times 10^6)} \\\\a = 4.38\times 10^{-15} \ m/s^2[/tex]

Thus, the change in mean drift velocity for electrons as they pass from one end of the wire to the other is 3.506 x 10⁻⁷ m/s and average acceleration of the electrons is 4.38 x 10⁻¹⁵ m/s².

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