Download SGBAU BSc 2019 Summer 6th Sem Mathematics Special Theory Of Relativity Question Paper

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B.Sc. Part-III (Semester-VI) Examination

MATHEMATICS

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(Special Theory of Relativity)

Paper—XII

Time : Three Hours] [Maximum Marks : 60

Note :—— (1) Question No. 1 is compulsory, attempt once.

(2) Attempt ONE question from each unit.

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1. Choose the correct alternative :

(i) The interval ds² = —(dx¹)² — (dx²)² — (dx³)² + (dx⁴)² is said to be space like if : 1

(a) ds²>0 (b) ds² <0

(c) ds²=0 (d) None of these

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(ii) The electric and magnetic field strengths E and H are invariant under : 1

(a) Galilean Transformations (b) Laplace Transformations

(c) Fourier Transformations (d) Gauge Transformations

(iii) A^µ =(A, φ)=(A_x,A_y,A_z,φ) is a four potential then : 1

(a) A = (A,φ) (b) A = (A,-φ)

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(c) A = (A,iφ) (d) A = (A,-iφ)

(iv) A^µ = (A¹, A², A³, A⁴) is a four vector or four dimensional vector where A² < 0 then A^µ is : 1

(a) Time like (b) Null or light like

(c) Space like (d) None of these

(v) Covariant tensor of rank one T_µ is defined as : 1

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(a) T_µ = (∂x^l/ ∂x^µ) T_l (b) T^l = (∂x^l/ ∂x^µ) T_µ

(c) T_µ = (∂x^µ/ ∂x^l) T_l (d) T_µ = (∂x^l/ ∂x^µ) T^l

(vi) The special Lorentz transformations will reduce to simple Galilean transformations when : 1

(a) V=C (b) C<<V

(c) V<<C (d) None of these

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(vii) If F^µν is defined as (electromagnetic field tensor) then F^µν is defined as : 1

(a) F_µν = (∂A_ν/ ∂x^µ) - (∂A_µ/ ∂x^ν) (b) F_µν = (∂A_µ/ ∂x^ν) - (∂A_ν/ ∂x^µ)

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(c) F_µν = (∂A_ν/ ∂x^µ) + (∂A_µ/ ∂x^ν) (d) None of these

(viii) The transformations x'=x- vt and t' = t are : 1

(a) Laplace transformations (b) Lorentz transformations

(c) Galilean transformations (d) None of these

(ix) If A is a vector potential then the magnetic field is : 1

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(a) H=div.A (b) H=Curl A

(c) E=div.(Curl A) (d) None of these

(x) Four velocity of a particle is : 1

(a) a unit space-like vector (b) a unit time-like vector

(c) a unit light-like vector (d) None of these

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UNIT—I

2. (a) Obtain Galilean transformation equations for two inertial frames in relative motion. 3

(b) Show that simultaneity is relative in special relativity. 3

(c) Show that the electromagnetic wave equation :

∂²/∂x² + ∂²/∂y² + ∂²/∂z² - (1/c²) ∂²/∂t² = 0

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is not invariant under the Galilean transformations. 4

3. (p) Discuss the geometrical interpretation of Lorentz transformations. 4

(q) Prove that ∇² - (1/c²)(∂²/∂t²) is invariant under special Lorentz transformations. 4

(r) Show that x² + y² + z² — c²t² is Lorentz invariant. 2

UNIT—II

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4. (a) Obtain the transformations for the velocity of a particle under special Lorentz transformations. 5

(b) If U and U' be the velocities of a particle in two inertial systems s and s' respectively where s' is moving with velocity v relative to s along the XX' axis then show that :

tan θ' = sinθ(1 - v²/c²)^1/2 / (cosθ - v/u)

and

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u' = (u²sin²θ + γ²(ucosθ - v)²)^1/2

where θ and θ' are the angles made by u and u' with the X-axis respectively. 5

5. (p) If u and u' be the velocities of a particle in two inertial systems s and s' respectively then prove that :

(1 - u'²/c²)^1/2 = (1 - v²/c²)^1/2(1 - u²/c²)^1/2 / (1 + u.v/c²)

where s' is moving with velocity v relative to s along XX' axis. 5

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(q) Show that in nature no signal can move with a velocity greater than the velocity of light relative to any inertial system. 5

UNIT—III

6. (a) Define time-like, space-like and light-like intervals for the space time geometry of special relativity. 3

(b) Define a four tensor of the second order. Prove that :

(i) T'^µν = a^µ_l a^ν_k T^lk and

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(ii) T'_µν = a^l_µ a^k_ν T_lk

7. (p) Define a four vector A^µ. Show that :

A_l = -A^l, A₂ = -A², A₃ = -A³, A₄ = A⁴

(q) Show that two events which are simultaneous in one inertial frame may not be at the same time if the interval between two events is space-like. 4

(r) Write the Lorentz transformations in index form. 2

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UNIT—IV

8. (a) Deduce Einstein’s mass energy equivalence relation. 5

(b) Define : Four velocity. Prove that the four velocity in component form can be expressed as :

u^µ = dx^µ/dτ = (u/√(1-u²/c²), c/√(1-u²/c²))

where U = (u_x, u_y, u_z) = velocity of the particle. 1+4

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9. (p) Define : Four momentum vector p^µ. Prove that the square of the magnitude of the four momentum vector p^µ is m²c². 1+4

(q) A particle is given a kinetic energy equal to n times its rest energy m₀c². Find speed and momentum of the particle. [ Kinetic energy=T=m₀c²(γ-1) ] 5

UNIT—V

10. (a) Show that the Hamiltonian for a charged particle moving in an electromagnetic field is :

H = mc² + c[ (p - eA/c)² + eφ ]^1/2 5

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(b) Define : Current four vector. Show that c²ρ² — J² is invariant and its value is ρ₀²c². 1+4

11. (p) Prove that the set of Maxwell's equations div.H=0 and Curl E + (1/c)(∂H/∂t) = 0 can be written as ∂F^µν/∂x^ν = 0 where F is the electro-magnetic field tensor. 5

(q) Define electromagnetic field tensor F^µν. Express the components of F^µν in terms of the electric and magnetic field strengths. 1+4

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