how do u prove that radius of curvature is twice the focal length in a concave mirror?

Question

prove R=2F in a concave mirror.
r = radius of curvature , f = focal length

Answer 1

You can't actually prove it. And it never is (in any practical sense). It's just close enough for physics.

Here is a theoretical sort of proof.

The focal point is the point at which parallel rays converge. The parallel rays will converge at a point that is reflected with incidence and reflected angles equal measured from the perpendicular (at the point of incidence)on the mirror's surface..

Draw it.

Answer 2

A ray of light passing parallel to the principal axis and close to the principal axis after incident at a point Q on the mirror is reflected according to the law of reflection ( i = r ) and passes through the focus F.

If O is the center of curvature of the mirror, consider the triangle OQF.

Angle oQf = r the angle of reflection.

Angle qOf = i the angle of incidence (the line OQ passes through two parallel lines). And therefore the Angle qOf = i

As i = r the triangle is a isosceles tri angle and hence QF = OF

If P is the pole of the mirror, considering large radius of curvature we can take PF =QF and hence OF = PF or half the radius of curvature.

Answer 3

You don't need to prove that. I never did, at least. All of my teachers never asked for it. It was proven by scientists before, so there is no point in proving it.

Answer 4

uhhmm, i dont know