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 does not depend on the cross-sectional area. It is a function of the length of the wire (L) and the time (t) rather than its cross-sectional area (A).

Step-by-step solution

Recall the formula for drift speed

The drift speed (v_d) of free electrons in a wire can be expressed as: v_d = I / (n * A * q), where I is the current flowing through the wire, n is the number density of the free electrons (number of electrons per unit volume), A is the cross-sectional area of the wire, and q is the charge of an electron. Our goal is to show that the drift speed does not depend on the cross-sectional area (A) of the wire.

Express current in terms of charge and number of electrons

We know that the current flowing through a wire (I) can be expressed as the charge (Q) passing through the wire per unit time (t): I = Q / t, where Q = n * V * q. Here, V represents the volume of the wire through which the charge passes.

Relate the volume of the wire with its cross-sectional area

We can express the volume of the wire (V) as a product of its cross-sectional area (A) and length (L): V = A * L.

Substitute Q and V in the current formula

Now we substitute Q = n * V * q and V = A * L in the current formula. I = (n * A * L * q) / t.

Simplify the formula for drift speed

By rearranging the equation, we can find the drift speed (v_d) of the free electrons in the wire. v_d = I / (n * A * q) = (n * A * L * q) / (t * n * A * q) = L / t.

Conclude that drift speed does not depend on the cross-sectional area

From the above formula, we see that drift speed (v_d) is given by the ratio of length (L) to time (t), and there is no mention of the cross-sectional area (A) in this expression. Thus, we can conclude that the drift speed of free electrons in a wire does not depend on its cross-sectional area.