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Total No. of Questions: 05
Total No. of Printed Pages : 05
RJ-194
B.Sc.-B.Ed. (Secondary) Examination, 2024
(Second Semester)
PHYSICS
DC-III
Mathematical Physics-II
Time : 3 Hours
Maximum Marks : 50
Note : Attempt all questions. All questions carry equal marks. There is internal choice in each question.
1.
(a) r makes equal angles α, β, γ with the coordinate axes. Prove that :
cos² α + cos² β + cos² γ = 1.
6
(b) If the system is in equilibrium, find tension, T in OA and OB.
6
2.
(a) Find the volume of a spherical shall of inner and outer radius r₁ and r₂ respectively using spherical coordinate system.
6
(b) Represent the vector A = zî - 2xj + yk in cylindrical coordinate system.
6
Or
P(x, y, z) and edges parallel to the coordinate axes and having magnitude Δx, respectively is given approximately by ∇ . (ΔV r).
Evaluate ∫∫ A.dS, where A = 18zî - 12j + 3yk and S is that part of the plan 2x+3y+6z=12, which is located in the first octant.
12
3.
The equation of motion of a particle is given by
m = f(r) r
d²r/dt² = f(r) r
where r is the position vector of P measured from an origin, o, f(r) is the force depending on r.
(a) Show that r x dr/dt = c (constant vector).
12
(b) Interpret the result physically and geometrically.
Or
Find the angle between the surfaces z = x² + y² and z = (x-√6/6)² + (y-√6/6)² at the point (√6/12, √6/12, 1/12).
12
4.
(a) A vector field F = x²î + y²ĵ + z²k is given, verify the divergence theorem for F in the region bounded by the sphere x² + y² + z² = R².
12
Or
For the magnetic field B = -yî + xĵ, verify Stoke's theorem for the plan z = 0, bounded by the circle x² + y² = 1.
12
5.
(a) State and prove Stoke's theorem.
12
Or
(b) Consider temperature distribution : T(x,y) = T₀ exp(-x² - y²).
find ∇²T and comment on the nature of the temperature field.
12
Or
Charge distribution in a region is given by ρ(x, y, z) = 3x + 4y - 2z. If the electric potential satisfies Poisson's equation ∇²φ = -ρ/ε₀.
12