Important Derivations: Electromagnetic Waves and Optical Instruments

Table of Contents
1. Important Derivations​
2. Derivation of Displacement Current
3. Derivation of Lens Formula
4. Derivation of Prism Formula
5. Derivation of Lens-maker Formula
6. Derivation of Mirror Formula
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Important Derivations
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Derivation of Displacement Current

The idea of displacement current arises when applying Ampère's circuital law to time-varying electric fields (for example, between the plates of a charging capacitor). Maxwell introduced the displacement current term to make Ampère's law consistent with the continuity equation for charge and with the conservation of charge.

Consider a parallel-plate capacitor with plate area A, plate separation d and electric field E(t) between the plates. Let the instantaneous charge on the plates be Q(t). The electric flux through the surface between the plates is

Φ_E = E·A

Because E = Q/(ε₀A) (in vacuum), the flux may be written as

Φ_E = Q/ε_o

Differentiate the flux with respect to time to relate it to the charging current ,i = dQ/dt.

dΦ_E/dt = (1/ε_o) dQ/dt = i/ε_o

Multiply both sides by ε0 to isolate the current term that must be present in Maxwell-Ampère law:

ε₀ (dΦ_E/dt) = i

Interpretation and Maxwell's correction:

When applying Ampère's law to a surface that spans the space between capacitor plates, the conduction current I flows onto the plates, but there is no conduction current through the surface bounded by the loop inside the dielectric gap. To preserve consistency of the law for any choice of surface bounded by the same closed curve, Maxwell introduced a term called the displacement current, Id, defined by

i_d = ε_o (dΦ_E/dt)

Thus the corrected (Maxwell-Ampère) law in integral form becomes

∮ B·dl = μ_o (i + i_d) = μ_oi + μ_o ε_o dΦ_E/dt

For a linear dielectric medium of permittivity ε, the generalised displacement current is

i_d = ε (dΦE/dt)

• Displacement current is not a flow of charges; it is a term representing the time rate of change of electric flux and ensures continuity of current in circuital laws.

Velocity of Propagation of an Electromagnetic Wave

Using Maxwell's equations in free space, one can derive the wave equation for electric and magnetic fields and obtain the speed of electromagnetic waves. Consider vacuum (no free charges or currents):

Maxwell's curl equations in vacuum are

∇×E = -∂B/∂t

∇×B = μ₀ ε₀ ∂E/∂t

Take the curl of the first equation and use the vector identity ∇×(∇×E) = ∇(∇·E) - ∇²E. In vacuum ∇·E = 0, so

∇×(∇×E) = -∇²E

Applying curl to ∇×E = -∂B/∂t gives

∇×(∇×E) = -∂(∇×B)/∂t

Substitute ∇×B from Maxwell's second equation:

-∇²E = -∂/∂t (μ₀ ε₀ ∂E/∂t)

Simplify to obtain the wave equation for E:

∇²E = μ₀ ε₀ ∂²E/∂t²

This is the standard wave equation with propagation speed v given by

v = 1/√(μ₀ ε₀)

Similarly, B satisfies the same form of wave equation and propagates at the same speed v.

Using known vacuum constants μ0 and ε0 gives v = c, the speed of light in vacuum.

Derivation of Lens Formula

What is the Lens Formula?

The lens formula relates the object distance u, the image distance v and the focal length f of a thin lens. For a thin lens (convex or concave) in air, the formula is

1/v - 1/u = 1/f

Using the sign convention commonly used in geometrical optics, this is usually written as

1/v + 1/u = 1/f

Derivation (thin convex lens, real inverted image)

Consider a thin convex lens with optical centre O and principal focus F. An object AB is placed perpendicular to the principal axis at distance u from O. A real, inverted image A'B' is formed at distance v on the other side of the lens.

From the given figure, we notice that △ABO and △A'B'O are similar.
Therefore,

Similarly, △A'B'F and △OCF are similar, hence

But,
OC = AB
Hence,

Equating eq (1) and (2), we get

Substituting the sign convention, we get
OB=-u,  OB'=v and OF=f

Dividing both the sides by uvf, we get

The above equation is known as the Lens formula.

Derivation of Prism Formula

Prism: 

A prism is a transparent optical element with flat, polished surfaces that refract light. Prisms are used for dispersion, reflection, beam steering, and polarisation control. Common types include:

• Dispersive prisms: break white light into constituent spectral colours (e.g., triangular prism).
• Reflective prisms: reflect and redirect beams (e.g., pentaprism, dove prism).
• Polarising prisms: split beams by polarisation (e.g., Nicol, Glan-Taylor).
• Beam-splitting prisms: divide beams into two or more (e.g., cube beam splitter).
• Deflecting prisms: deviate beams by a fixed angle (e.g., wedge prism).

Derivation of the Prism Formula (small-angle approximation)

Consider a thin triangular prism of small apex angle A and refractive index μ. A monochromatic ray incident at small angle i is refracted twice and emerges with overall deviation δ. Using Snell's law at both faces and under the paraxial/small-angle approximation (sin θ ≈ θ, for θ in radians), one obtains a simple relation between δ, A and μ.

From geometry inside the prism the angle of refraction r at the first face and r' at the second face satisfy

r + r' = A

Thus, AL = LM and LM ∥ BC

Thus, above is the prism formula.

Derivation of Lens-maker Formula

The lens maker's formula gives the focal length f of a thin lens in terms of its refractive index μ and the radii of curvature R1 and R2 of its two spherical surfaces. Lens manufacturers use this formula to design lenses of desired focal lengths.

Assumptions

• The lens is thin so that thickness can be neglected compared to object and image distances.
• The surrounding medium on both sides of the lens is the same (usually air) with refractive index n1 (≈1).
• Radii of curvature of the two surfaces are R1 and R2, with sign convention: convex surface towards incoming light has positive radius, concave negative, consistent with the chosen optical sign convention.
• Paraxial approximation (small angles) is used.

Derivation

The complete derivation of the lens maker formula is described below. Using the formula for refraction at a single spherical surface, we can say that,
For the first surface,

For the second surface,

Now adding equation (1) and (2),

When u = ∞ and v = f

But also,

Therefore, we can say that,

Where μ is the refractive index of the material.

Limitations of the Lens-maker Formula

• The lens must be thin; for thick lenses the separation between surfaces cannot be neglected and a more general thick-lens formula must be used.
• The medium on both sides of the lens must be the same; if not, a modified form with the two medium refractive indices is required.
• Paraxial approximation is implicit; large-angle rays will deviate and spherical aberration must be considered.

Derivation of Mirror Formula

Mirror formula and assumptions

The mirror formula for a spherical mirror relates object distance u, image distance v and focal length f via

1/u + 1/v = 1/f

Assumptions and sign conventions used in the derivation:

• Distances are measured from the pole P of the spherical mirror.
• The direction of incident light is taken as positive; distances measured opposite to this direction are negative (conventional sign rules may vary-be consistent throughout).
• Paraxial approximation is used so that small-angle geometry and similar triangles are valid.

Derivation (concave spherical mirror)

Consider a concave spherical mirror with centre of curvature C, pole P and focal point F. An object AB is placed at distance u from P and forms an image A1B1 at distance v from P.

From the figure given above, it is obvious that the object AB is placed at a distance of u from P which is the pole of the mirror. From the diagram we can also say that the image A₁B₁ is formed at vfrom the mirror.
Now from the above diagram, it is clear that according to the law of vertically opposite angles the opposite angles are equal. So we can write:
∠ACB = ∠A₁CB₁;
Similarly;
∠ABC=∠A₁B₁C; (right angles)
Now since two angles of triangle ACB and A₁CB₁ are equal and hence the third angle is also equal and is given by;
∠BAC = ∠B₁A₁C; and

Similarly the triangle of FED and FA₁B₁ are also equal and similar, so;

Also since ED is equal to AB so we have;

Combining 1 and 2 we have;

Consider that the point D is very close to P and hence EF = PF, so;

From the above diagram BC = PC - PB and B₁C = PB₁ - PC and FB₁ = PB₁ - PF;

Now substituting the values of above segments along with the sign, we have;
PC = -R;
PB = u;
PB₁ = -V;
PF = -f;
So the above equation becomes;

Solving it we have;
uv - uf - Rv + Rf = Rf - vf;
uv - uf - Rv + vf = 0;
since R = 2f (radius of curvature is twice that of focal length), hence;
uv - uf -2fv + vf = 0;
uv - uf - vf = 0;
Solving it further and dividing with "uv" we have;

Remarks on sign convention and applications

• Use a consistent sign convention (for example, the Cartesian sign convention: object distances measured to the left of pole are negative, image distances to the right positive, radii positive if centre is to the right of pole, etc.).
• The mirror formula is routinely used to locate images for object positions and to find focal lengths experimentally.