Prism
A transparent refracting medium bounded by two plane surfaces meeting each other along a straight edge is called a prism.
The two inclined plane surfaces through which light passes are called refracting faces of the prism. The angle between two refracting faces is called angle of prism (A).

Refraction through a prism / Minimum deviation and their Relation
We consider, a ray of light ST is incident at a point ‘T’ on the surface PQ of a prism PQR by making an angle of incidence ‘i’ and refracted along TU by making angle of refraction ‘r₁‘. Again, the refracted ray TU is incident at point U at angle of incidence ‘r₂‘ and refracted along UV at angle of emergence ‘e’.

Let, NO and N’O be the normal at the faces PQ and PR at the points T and U respectively.
From figure, ∠QPR = A = angle of prism, ∠WTU = [latex]\alpha[/latex], ∠WUT = [latex]\beta[/latex] and ∠XWU = [latex]\delta[/latex].
From the figure,
[latex]\therefore i = \alpha + r_1[/latex] …………….. (i)
And, e = [latex]\beta + r_2[/latex] ………………… (ii)
From eq^ns. (i) and (ii), we get,
i + e = [latex]\alpha + \beta + r_1 + r_2[/latex]
i + e = [latex]\delta + r_1 + r_2[/latex] ………………… (iii)         [latex][\because \alpha + \beta = \delta][/latex]
From quadrilateral, PTOU,
A + 90 + r + 90 = 360^o
Or, A + r = 180^o
Also, from [latex]\Delta[/latex]TOU, r + r₁ + r₂ = 180^o
[latex]\therefore[/latex] A + r = r + r₁ + r₂
Or, A = r₁ + r₂ ………………… (iv)
So, from eq^ns. (iii) and (iv), we get,
i + e =  ………….. (v)
This relation shows that the sum of angle of incidence and angle of emergence is equal to the sum of the angle of deviation and angle of prism.

Minimum deviation
It is defined as the minimum angle of deviation. When we plot the graph between angle of incidence and deviation, then, we get a curve as shown in figure. It is represented by the symbol [latex]\delta_m[/latex]

When i = e and r₁ = r₂ = r, then, the deviation is minimum,
i + e = A + [latex]\delta[/latex]
Or, i + i = A + [latex]\delta_m[/latex]
Or, 2i = A + [latex]\delta_m[/latex]
Or, i = [latex]\frac{A+\delta_m}{2}[/latex] ……………. (vi)
Again, r₁ + r₂ = A,
Or, r + r = A
Or, 2r = A
Or, r = [latex]\frac{A}{2}[/latex] ………….. (vii)

Therefore, [latex]\mu = \frac{sini}{sinr}[/latex]

[latex]\therefore \mu = \frac{sin(\frac{A+\delta_m}{2})}{sin(\frac{A}{2})}[/latex] ……………… (viii)
This is the required relation between the refractive index [latex](\mu)[/latex], angle of prism (A) and minimum deviation [latex](\delta_m)[/latex].

Deviation through a small angled prism
The prism whose angle lies between 6^o – 12^o is known as small angled prism.

Let, XYZ be the prism whose angle is A. PQ is incident ray incident at point Q and refracted along QR by making angle of incident ‘i’ and angle of refraction r₁.
The refractive index is given by:
[latex]\mu = \frac{sini}{sinr}[/latex] ……………. (i)
When the prism is of small angle, then, Sini ≈ i, Sinr ≈ r.
[latex]\therefore \mu = \frac{i}{r_1}[/latex]
Or, i = [latex]\mu .r_1[/latex] ………………………….. (ii)
The incident ray SR is incident at R and refracted along RQ by making angle of incidence R and angle of refraction r₂. The direction of ray is taken as reverse.
[latex]\therefore \mu = \frac{sine}{sinr_2}[/latex] …………………… (iii)
When the prism is small, the, Sine ≈ e and Sinr₂ ≈ r₂.
[latex]\therefore \mu = \frac{e}{r_2}[/latex]
e = [latex]\mu .r_2[/latex] …………… (iv)
Again, i + e = [latex]\delta + A[/latex]
Or, [latex]\delta = i + e – A = \mu r_1 + \mu r_2 – A[/latex]
= [latex]\mu(r_1 + r_2) – A[/latex]
            = [latex]\mu A – A[/latex]
[latex]\delta = (\mu – 1)A[/latex]
This relation gives the deviation due to small angled prism.
Conclusions

1. The deviation angle is dependent with the angle of prism and refractive index of the material of the prism.
2. The deviation angle is independent with the angle of incidence.

Grazing Incidence and Grazing Emergence

When a ray of light is incident on a prism by an angle of incidence 90^o, the ray lies on the surface and refracted through prism. This refraction of the prism is called grazing incidence.
From fig. (a),
∠NQP = 90^o, angle of refraction ∠MQR = critical angle (c).
In this case,
Maximum angle of deviation,
[latex]\delta_{max} = (i – r_1) + (e – r_2)[/latex]
            = 90^o – c + e – r
            = 90^o + e – (c+r)
When a ray of light PQ is at grazing incidence on the surface XY of a prism XYZ and angle of prism is gradually increased, the refracted ray QR on the face XZ will make bigger and bigger angle of incidence. In this case, the emergent ray will graze along the surface XZ, as shown in figure (c). So, we have,
∠QRT = critical angle (c)
So, fig. (c) shows the largest angle of prism for grazing emergence. This angle is called limiting angle of prism. So, the limiting angle,
A = c + c = 2c.             [∵ A = r₁ + r₂]
Hence, the limiting angle of prism is twice the critical angle.