# Optionals IAS Mains Physics: Questions 255 - 260 of 307

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## Question 255

Appeared in Year: *2011*

### Describe in Detail

Essay▾Let be the vector operator with components equal to the Pauli spin matrices and . If and are the vectors in 3 – dimensional space, prove the identity

### Explanation

L. H. S

Using identities:

(Identity)

Proved

L. H. S R. H. S

## Question 256

Appeared in Year: *2009*

### Describe in Detail

Essay▾Obtain an expression for the resolving power of a plane transmission grating. (Paper-II Marks - 15)

### Explanation

- The resolving power of a grating is defined as the ratio of the wavelength of any spectral line to the smallest difference in wavelength , between this line and a neighbouring line such that the two lines appear just resolved, according to Rayleigh՚s criterion. So, resolving power of a grating
- In fig 1, XY is the grating surface and MN is the fie…

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## Question 257

Appeared in Year: *2009*

### Describe in Detail

Essay▾Derive the condition for achromatism of two thin lenses separated by a finite distance and made up of same material. (Paper I Marks - 10)

### Explanation

If the two lenses are of same material, their achromatic combination is possible only if they are separated by a finite distance.

Let be the focal lengths of the two lenses placed coaxial and separated by a distance (fig 1) . This mean focal length F of the combination is given by

Differentiating the above equation, we obtain

Bu…

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## Question 258

Appeared in Year: *2009*

### Describe in Detail

Essay▾Show that a four – dimensional volume elements is invariant to Lorentz transformation.

(Marks: 10)

### Explanation

According to Lorentz Fitzgerald Contraction, we have

∴ Three dimensional volume element in system

=

According to time dilation

∴ Four dimensional volume element in system

(i.e.. four dimensional volume element system )

Hence four-dimensional volume element is invariant under Lorentz transformations.

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## Question 259

Appeared in Year: *2009*

### Describe in Detail

Essay▾Show that the plane of polarisation is rotated through in optical rotation where symbols have their usual meanings. (Paper-II Marks - 15)

### Explanation

Let us assume that linearly polarised light is incident normally on a quartz plate cut perpendicular to the optic axis. Let the vibrations in the incident light be represented by

.

On entering the crystal these vibrations are broken up into two equal and opposite circular vibrations which are represented by

These circular components travel though the …

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## Question 260

Appeared in Year: *2009*

### Describe in Detail

Essay▾Consider a spherical shell of mass M & radius R. Calculate the potential due to this shell at a point P when the point is (i) outside the shell and (ii) inside the shell

If the spherical shell is now replaced by a uniform solid sphere of same mass and radius, what will be its potential at the same external point? (Paper-II Marks - 15)

### Explanation

**At a point outside the shell**:

- Let P be a point at a distance r form the centre O of the spherical shell of radius R (fig 1) and surface density (i.e. mass per unit area of surface ) .
- Join OP and cut out a slice CEFD in the form of a ring by two planes CD and EF close to each and perpendicular to the radius OA of the shell and let angle EOP be Q, the…

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