Question

A circuit consists of a battery, whose emf is K, and five Nichrome wires, three thick and two thin as shown in Figure 19.78. The thicknesses of the wires have been exaggerated in order to give you room to draw inside the wires. The internal resistance of the battery is negligible compared to the resistance of the wires. The voltmeter is not attached until part (e) of the problem. (a) Draw and label appropriately the electric field at the locations marked × inside the wires, paying attention to appropriate relative magnitudes of the vectors that you draw. (b) Show the approximate distribution of charges for this circuit. Make the important aspects of the charge distribution very clear in your drawing, supplementing your diagram if necessary with very brief written descriptions on the diagram. Make sure that parts (a) and (b) of this problem are consistent with each other. (c) Assume that you know the mobile-electron density n and the electron mobility u at room temperature for Nichrome. The lengths $(L_1,L_2,L_3)$and diameters $(d_1,d_2)$of the wires are given on the diagram. Calculate accurately the number of electrons that leave the negative end of the battery every second. Assume that no part of the circuit gets very hot. Express your result in terms of the given quantities $(K,L_1,L_2,L_3,d_1,d_2,n\text{and}u)$. Explain your work and identify the principles you are using. (d) In the case that ${\mathit{d}}_{\mathbf{2}}\mathbf{\ll }{\mathit{d}}_{\mathbf{1}}$, what is the approximate number of electrons that leave the negative end of every second? (e) A voltmeter is attached to the circuit with its + lead connected to location B (halfway along the leftmost thick wire) and its - lead connected to location C (halfway along the leftmost thin wire). In the case that ${\mathit{d}}_{\mathbf{2}}\mathbf{\ll }{\mathit{d}}_{\mathbf{1}}$, what is the approximate voltage shown on the voltmeter, including sign? Express your result in terms of the given quantities$(K,L_1,L_2,L_3,d_1,d_2,n\text{and}u)$.

Short answer

(b) The charges will be distributed as a small gradient along the thick wires, and the charges will be distributed as a larger gradient along the thin wires.

Step-by-step solution

Given data

A circuit with has a battery whose emf value is K, and the five Nichrome wires are connected. Out of 5, three are thick wires, and two are thin wires.

Concept

The number of charge carriers that are moving in a conductor which is connected to a circuit can be written as,

${\mathit{n}}_{\mathbf{e}}\mathbf{=}\mathit{n}\mathit{A}\mathit{u}\mathit{E}\mathit{t}$ (1)

Here n is the charge density of the charge carrier, A is the cross-sectional area of the conductor, E is the electric field in the conductor, u is the mobility of the charge carriers, t is the time.

(b) The approximate distribution of charges for this circuit 

The expression that describes the relationship between cross-sectional area and the electric field is,

$A_1{E}_1=A_2{E}_2$(Refer to SID: 875865-19-59 P-a)

Here A₁ is the cross-sectional area of the thick wire, E₁ is the electric field in the thick wire, A₂is the cross-sectional area of the thin wire, E₂is the electric field in the thin wire.

We know the thick wires have a smaller electric field; thus, the charges will be distributed as a small gradient along the thick wires.

For thin wires, the electric field is large; thus, the charges will be distributed as a larger gradient along the thin wires.

The central (mid) thick wire in the diagram has less charge by symmetry.

The distribution of the charges in the wires can be shown as,

Thus, the charges will be distributed as a small gradient along the thick wires, and the charges will be distributed as a larger gradient along the thin wires.