Introduction to Electricity and Magnetism B38EM
Assignment 1
0 = 8.85x10-12 Fm-1, e = 1.6x10-19 C, 1 nC = 10-9 C
1) Gradient of scalar function: Find the gradient of the following scalar function and then evaluate it
at the given point.
V1 = 24V0 cos (π y/3) sin (2π z/3)
at (3, 2, l) in Cartesian coordinates.
(3 marks)
2) Calculating the Divergence: Determine the divergence of the following vector field and
then evaluate it at the indicated point:
E = x^ 3x2 + y^ 2z + z^ x2z
at (2, −2, 0) in Cartesian coordinates.
(3 marks)
3) Circulation: Given that F = x2ax − xzay − y2az, calculate the circulation of F around the (closed)
path shown in the Figure.
(3 marks)
4) In a certain region, the electric flux density is given by:
D = 2ρ(z+1)cosφ aρ – ρ(z+1)sinφ aφ + ρ2 cosφ az μC/m2.
a) Show that the charge density is equal to ρv = 3(z+1)cosφ μC/m2
b) Calculate the total charge enclosed by the volume 0<ρ<2, 0<φ<π/2, 0<z<4.
(5 marks)
5) A spherical shell with outer radius b surrounds a charge-free cavity of radius a < b (Fig. 2). If the
shell contains a charge density given by
𝜌𝑉 = −
𝜌𝑉0
,
𝑅2
𝑎 ≤ 𝑅 ≤ 𝑏,
where ρV0 is a positive constant, determine the electric flux density D in all regions.
Fig. 2 for problem 5.
(5 marks)
6) Spherical Capacitor: Show that the capacitance of two concentric spherical metal shells with radii a
and b is equal to C = 4πε0[ab/(b−a)]. Assume the inner shell has charge +Q and the outer -Q.
(3 marks)
7) Consider a straight non-magnetic conductor of circular cross-section and radius a carrying a current with
uniform current density J (A/m2) in the vertical direction. Using Ampere’s law find the magnetic field inside and
outside the conductor.
(3 marks)
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hepotentialdifferencebetweentheinnerandoutersurfaces is
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