Question

Show that the drift speed of free electrons in a wire does not depend on the cross-sectional area of the wire.

Short answer

Answer: No, the drift speed of free electrons in a wire does not depend on the cross-sectional area of the wire.

Step-by-step solution

Understand the drift speed concept

Drift speed is the average velocity of free electrons in a conducting wire due to an electric field. When a voltage is applied across a wire, the electric field inside the wire exerts a force on the free electrons that causes them to move. The average speed of these electrons is known as the drift speed.

Expressing the drift speed in terms of current and number of electrons

The drift speed (\(v_d\)) can be expressed in terms of the current flowing through the wire (\(I\)), the number of free electrons in the wire (\(n\)), the charge of an electron (\(e\)), and the cross-sectional area of the wire (\(A\)). The formula is given by: \(v_d = \frac{I}{nAe}\)

Find the relationship between the current, cross-sectional area, and drift speed

According to Ohm's Law, the current (\(I\)) flowing through the wire is proportional to the voltage (\(V\)) applied across the wire and inversely proportional to the resistance (\(R\)) of the wire: \(I = \frac{V}{R}\) And the resistance (\(R\)) of a uniform wire is given by: \(R = \frac{\rho L}{A}\) Where \(\rho\) is the resistivity of the material, and \(L\) is the length of the wire.

Substitute the relationship of current and resistance into the drift speed formula

We can now substitute the expression for the current found in step 3 into the drift speed formula from step 2: \(v_d = \frac{\frac{V}{\frac{\rho L}{A}}}{nAe}\) Simplifying the equation, we get: \(v_d = \frac{VAe}{\rho L nAe}\)

Show drift speed is independent of the cross-sectional area

Now, we can see that the cross-sectional area (\(A\)) appears in both the numerator and the denominator of the equation for drift speed: \(v_d = \frac{V \cancel{A} e}{\rho L n \cancel{A} e}\) As the cross-sectional area cancels out, we are left with an expression for drift speed that does not depend on the cross-sectional area: \(v_d = \frac{V}{\rho L n e}\) Thus, we have shown that the drift speed of free electrons in a wire does not depend on the cross-sectional area of the wire.