Let XY be a plane surface separating air from a denser medium and PA be a plane wavefront just incident on it. The normals LA and MP to the incident wavefront represent incident rays. If AN is normal to the refracting surface at point A, then `angle NAL` = i is the angle of incidence as shown in FIg.
First of all, the wavefront reaches point A. From the points located between points A to P, the wavefront will reach from A to P later in time but in the same order, so that it will reach point P in the last.
Thus, different points on teh surface XY will become sources of secondary wavelents at different instans of time. When the distrubances from point P on incident wavefront jiust reaches the point (inside the denser medium) of radius AA. such that
`(A A)/(v) = (P P)/(c )`
or `A A. = (v)/(c ) xx P P`
where ve are the velocites of light in denser medium and air respectively. To find the refracted wavefront (new position of the wave front after reflection) , with point
A as centre, draw a sphere a radius A A = `(v)/( c ) xx P P` From point, P draw a tangent plane P A. to the sphere. Then P A. represents refracted wavefront and lines A A.L and P M normals to the refracted wavefront represent refracted rays. Also `angle N A A.j = r` is the angle of refraction.
To prove the laws of refraction : Consider any point Q on the incident wavefront. Suppose that when distrubance from point P on incident point Q via reaches point P on the refracted wavefrone, fhte disttubance from point, P A. represents the refracted wavefront, the time taken by light to travel from a point on incident wavefront to the corresponding point on reflected wavefront should be constant, Now time taken by light to go from Q to Will be
`t = (QK)/(c) + (KQ)/(v)`
In right angled `Delta AQK (QK)/(AK) = sin i`
Therefore, QK = AK sin i
Also, in right angled `Delta PQK, angel QPK = r`.
and `(KQ)/(AK) = sin r`
Therefore, `(AK)/(KO) = KP sin r`
In equation (i), substituting for QK and KQ, we have
`t = (AK sin i)/(c) + (KP sin r)/(v)`
`t = (AK sin i)/(c) + ((AP - AK)sin r)/(v) (therefore KP = AP - AK)`
or `t = (AP)/(v) sin r + AK ((sin i)/(c) - (sin r)/(v))`
From rays from different points on the incident wavefront, the values of AK are different, The rays from different points on incident wavefront will take the same time to reach the corresponding points on the refracted wavefront, if t, given by equation (ii), is independent of AK. It will happen so, if
`(sin i)/(c) - (sin r)/(v) = 0 or (sin i)/(sin r) = (c)/(v)`
But `(c)/(v) = mu` (constant) is called the refractive of the denser medium w.r.t air.
`therefore (sin i)/(sin r) = mu` (constant)
i.e., when a ray of light is incident on a refracting the ratio of the sine of the angle of incidence to the sine of the given pair of madia, which is Snel..s law of refraction.
Further, as explained in last section, it follows that the incidence ray (LA or MP), the normal (AN) and the refacted ray (A A. L or P M) all lie in the same plane.
Hence the laws of refraction are provied on the basis of the wave theory.
