Explain the physical mechanism that releases (10^61) ergs from an AGN (Active Galactic Nucleus) over its lifetime, given that the Schwarzschild radius of a (10^8 M_\odot) black hole is only ~2 AU.
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Question
Solution
Discussion Question 2
A point charge (q) is placed at a distance (d) from the center of an uncharged conducting sphere of radius (R) ((R < d)). Find the force on (q) due to induced charges. Discuss what happens if (q) is slowly moved closer to the sphere. Explain the physical mechanism that releases (10^61) ergs
Solution
Use method of images: Replace sphere by image charge (q' = -\fracRd q) at distance (b = \fracR^2d) from center on line joining center to (q).
Force on (q) due to induced charges = force due to (q'):
[
F = \frac14\pi\epsilon_0 \fracq q'(d - b)^2 = \frac14\pi\epsilon_0 \fracq\left(-\fracRd q\right)\left(d - \fracR^2d\right)^2
]
Simplify:
[
F = -\frac14\pi\epsilon_0 \fracq^2 R d(d^2 - R^2)^2
]
Negative ⇒ attractive. 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.