Test2017 sem1 - Class Test Paper 2017 PDF

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Class Test Paper 2017...

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PHY-20006 Semester 1 Maths Test: Answer all 4 questions. Time allowed 75 minutes. The Mathematics Handbook is available and some useful formulae can be found at the end of the paper. 1. An electric potential is given by V = e−2x + e−2y . (i) Show that the electric field is given by E = 2e−2x ˆi + 2e−2y ˆj

[3]

R (ii) The work done on a charge in an electric field is q E · dl. Calculate the work done in moving a charge of 2 Coulombs from the origin to the point (1,1) along the line y = x2 and show that this is equal to the change in potential energy. [5] (iii) Calculate the curl of the E-field and hence explain what is the work done in moving the charge back to the origin along any path. [4] 2. An electric field is given in spherical polar co-ordinates by E=

sin θ 2 cos θ ˆ r+ 3 ˆ θ 3 r r

in units of V m−1 . (i) Calculate the divergence of this electric field.

[4]

(ii) Hence or otherwise, determine the surface integral of the electric field emerging through a sphere of radius 2 m, centred on the origin. [4] 3. A cylinder of radius 1 m is aligned with the z-axis and has its flat faces at z = 0 and z = 10 m respectively. The cylinder is filled with a fluid of density (in kg m−3 ) given by the expression ρ(R, φ, z ) = 10 exp(−z) cos2 φ (i) Determine the total mass of fluid in the cylinder.

[6]

The bottom of the cylinder is removed and the fluid flows out at a constant rate (in kg m−2 s−1 ) determined by the vector field ˆ − 2R ˆ z. ρv = cos(φ) φ (ii) By performing a surface integral, determine how long it takes the fluid to drain from the cylinder. [6] Cont’d over

4. Poisson’s law in electrostatics can be written as ∇2 V = −ρ/ǫ0 , where V is the electric potential and ρ is a charge density in C m−3 . The electric potential due to a spherically symmetric charge distribution is given by

5 ǫ0 r 5r V (r) = ǫ0 V (r) =

for r > 1 for r ≤ 1

(i) Show that ρ = 0 for r > 1.

[3]

(ii) Determine an expression for ρ at r ≤ 1 and hence determine the total charge present. [5]

Useful Formulae The Laplacian in spherical coordinates is given as:1 ∂ ∇ Φ= 2 r ∂r 2

    ∂ 2Φ 1 ∂Φ 1 ∂ 2 ∂Φ r + 2 sin θ + 2 2 r sin θ ∂θ ∂r ∂θ r sin θ ∂φ2

Divergence of a vector in spherical polar coordinates:∇·A =

1 ∂ 1 ∂Aφ 1 ∂ 2 (r Ar ) + (Aθ sin θ) + 2 r sin θ ∂θ r sin θ ∂φ r ∂r

Element of volume in cylindrical polar coordinates:dV = R dR dφ dz Element of a flat surface in cylindrical coordinates:dS = ±R dR dφ ˆ z Element of volume in spherical polar coordinates:dV = r 2 sin θ dr dθ dφ Element of surface area in spherical polar coordinates:dS = r 2 sin θ dθ dφ ˆ r...