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A point charge lies on the axis of a circle of radius R, at a distance r from the center such that R = r*sqrt(3). If the electric flux passing through the circle is phi, what is the magnitude of the point charge?
- sqrt(3)*eps0*phi
- 2*eps0*phi
- 4*eps0*phi/sqrt(3)
- 4*eps0*phi
Correct answer: 2*eps0*phi
Solution
The point charge q is at distance r from center on axis. R = r*sqrt(3). Half-angle of cone: tan(theta) = R/r = sqrt(3) => theta = 60 deg. Solid angle: Omega = 2*pi*(1-cos(60 deg)) = 2*pi*(1-1/2) = pi sr. Flux through circle = q*Omega/(4*pi*eps0) = q*pi/(4*pi*eps0) = q/(4*eps0). Set equal to phi: q/(4*eps0) = phi => q = 4*eps0*phi. But option D is 4*eps0*phi... Wait, let me re-examine: The problem says R = a*sqrt(3) and r is the distance (using variable 'a' for the axis distance? or 'r' = a). If R = a*sqrt(3) and the distance is 'r' then R = r*sqrt(3) means r is the axis distance. Half-angle: tan(theta)=R/r=sqrt(3) => theta=60 deg. Omega = 2pi(1-cos60)=2pi(0.5)=pi. phi = q*Omega/(4pi*eps0) = q/(4*eps0). So q = 4*eps0*phi. Answer: 4*eps0*phi (option D). But wait, checking option B: 2*eps0*phi means Omega=2pi/... Let me re-examine if the charge is on the other side: same result. Answer should be 4*eps0*phi.
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