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I'm trying to follow the original Blandford-Znajek paper (found here), and I can't find how equations 2.1, 2.2 and 2.3 are obtained:

2.1 refers to "the charge density required to ensure that $E \cdot B = 0$":

$$ \rho \approx \epsilon_0 \frac{a}{M^2} c B $$

2.2 refers to "potential difference across gap of height $h$ when no charge is present":

$$ \Delta V \approx \frac{a}{M^2} c B h^2 $$

2.3 refers to "energies of produced pairs in the unstable vacuum":

$$ \gamma \approx \Big ( \frac{h}{M} \Big )^2 \Big ( \frac{ \omega_G a }{c} \Big ) $$

the relevant page is this:

relevant page

There is some mention to a voltage over a gap $h$, a current $j$ and charge density $\rho$, but is not clear what is happening and how everything fits together. There is neither a reference to another paper or textbook where that formula is obtained, so I presume that their derivation should be really trivial and easy, but I can't figure out what that is. I look the figures in the paper, but none that clearly refers to this argument

That section also talks about how the magnetosphere is generated (see image below). Note this fragment:

The currents that pervade the magnetosphere as sources of the magnetic field are presumably carried by charged particles that are flowing outwards at large distances

This assertion seems very confusing to me. Why "outwards"? isn't the magnetosphere caused by infalling accretion matter? why would anything be moving "outwards" at all?

Does anyone understand what is the geometry being described?

source of the magnetosphere

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  • $\begingroup$ I just glanced at the Ruderman and Sutherland paper cited in the second paragraph of section 2. It seems to mention $E\cdot B$ a lot... I would look there: adsabs.harvard.edu/abs/1975ApJ...196...51R $\endgroup$ – Paul T. Jun 23 '15 at 18:27
  • $\begingroup$ @PaulT. looking at section II of that paper, they seem to make the assumption that most of the magnetic field comes from the neutron star degenerated core. I think that assumption cannot be extrapolated as is for black holes. I wonder why Blandford/Znajek quote it as if it was relevant without further details $\endgroup$ – lurscher Jun 24 '15 at 1:46

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