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Physics Galaxy Discussion Questions Solutions Direct

Physics Galaxy Discussion Questions Solutions Direct

Introduction

Solution

Discussion:
As (d \to R^+), (F \to -\infty) (image charge (q') increases and distance to image decreases). The force becomes very strong — but before touching, the point charge induces large surface charge density, leading to possible discharge or breakdown. If the sphere is grounded, then image charge (q' = -\fracRd q) still holds but total charge on sphere is not zero.

Physics Galaxy Insight:

Mathematical check:
For flat $v(r) = v_0$, enclosed mass $M(r) = \fracv_0^2 rG$. Differential rotation curve slope: $\fracdvdr = 0$ implies $\fracdMdr = \fracv_0^2G$, so $\rho(r) = \frac14\pi r^2 \fracdMdr = \fracv_0^24\pi G r^2$.

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Introduction

Solution

Discussion:
As (d \to R^+), (F \to -\infty) (image charge (q') increases and distance to image decreases). The force becomes very strong — but before touching, the point charge induces large surface charge density, leading to possible discharge or breakdown. If the sphere is grounded, then image charge (q' = -\fracRd q) still holds but total charge on sphere is not zero.

Physics Galaxy Insight:

Mathematical check:
For flat $v(r) = v_0$, enclosed mass $M(r) = \fracv_0^2 rG$. Differential rotation curve slope: $\fracdvdr = 0$ implies $\fracdMdr = \fracv_0^2G$, so $\rho(r) = \frac14\pi r^2 \fracdMdr = \fracv_0^24\pi G r^2$.

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