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Question #42699

Difficulty: 8/10 | 🟢 Free

The Question

An infinite straight wire carrying a uniform linear charge density $\lambda$ is placed along the $z$ -axis. A sequence of small conducting spheres, numbered $n = 0, 1, 2, \dots$ , each of radius $R$ , are placed along the $x$ -axis at positions $x_n = a \cdot 2^n$ (where $R \ll a$ ). Adjacent spheres $n$ and $n+1$ are connected by a thin wire with a lumped capacitor of capacitance $C$ . The self-capacitance of each sphere is $C_0 = 4\pi\epsilon_0 R$ . Given that $C_0 = \frac{4}{3}C$ , and the system is assembled such that the net charge on each isolated section (i.e., sphere $n$ and the two capacitor plates directly connected to it form an isolated neutral island) is zero, find the steady-state charge $Q_0$ on the sphere at $n=0$ .

Options

A.
$-R \lambda \ln 2$
B.
$+R \lambda \ln 2$
C.
$-\frac{4}{3} R \lambda \ln 2$
D.
$+\frac{3}{4} R \lambda \ln 2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Infinite Series Capacitance|Infinite Wire vs. Finite Wire|Vector Integration

Why This Question Is Tricky

Common Mistakes

This problem tests the superposition of a non-uniform external electric field on an infinite capacitor network.

A common trap is assuming the standard ladder network equations apply directly to the total node potentials, ignoring that the external field biases the self-charge of each node ( $Q_n = C_0(U_n - V_{ext})$ ).

Students might also attempt to sum the induced charges directly, leading to a divergent series.

The key physical insight is recognizing that the logarithmic potential of the infinite wire, when sampled at geometric distances, creates a linear potential bias in the node index $n$ .

By defining the potential deviation $\delta_n = U_n - V_{ext}(x_n)$ , the non-homogeneous recurrence relation beautifully collapses into a homogeneous one, allowing a clean analytical solution.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Infinite Series Capacitance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free