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PHY31M1: Electromagnetism Test 1
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Instructions
1) Answer all questions.
Useful Hints
Transformation of vector 𝐴⃗ = (𝐴𝑥 , 𝐴𝑦 , 𝐴𝑧 ) in cartesian to cylindrical (𝐴𝜌 , 𝐴𝜑 , 𝐴𝑧 ) is given by.
𝐴𝜌
𝑐𝑜𝑠𝜑
(𝐴𝜑 ) = (−𝑠𝑖𝑛𝜑
0
𝐴𝑧
𝑠𝑖𝑛𝜑
𝑐𝑜𝑠𝜑
0
0 𝐴𝑥
0) (𝐴𝑦 )
1 𝐴𝑧
And
𝐴⃗ = (𝐴𝑥 , 𝐴𝑦 , 𝐴𝑧 ) in spherical is given by
𝐴𝑟
𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃
( 𝐴𝜃 ) = (𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜑
𝐴𝜑
−𝑠𝑖𝑛𝜑
𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜑
𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜑
𝑐𝑜𝑠𝜑
𝐴𝑥
𝑐𝑜𝑠𝜃
−𝑠𝑖𝑛𝜃) (𝐴𝑦 )
0
𝐴𝑧
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1) Write
(i)
(ii)
(iii)
(iv)
(v)
short notes on the following
Gradient
Divergence
Curl
Stokes’s theorem
Divergence theorem
⃗⃗ = 𝑦𝑎̂𝑥 + (𝑥 + 𝑧)𝑎̂𝑦 and Q is located at (-2.6,3)
2) If 𝐵
(i)
Express the point Q in
(a) Cylindrical coordinates
(b) Spherical coordinates
[2]
[2]
[3]
[4]
[4]
[5]
[5]
3) An electric field points in the z-direction everywhere in space. The magnitude
of the electric field depends linearly on the x-position in space, so that the
electric field vector is given by 𝐸⃗⃗ = (1 − 2𝑥)𝑎̂𝑧 . Calculate the flux of the
electric field through a square of side 4 cm that is in the positive xy plane by
following the steps below.
(i)
(ii)
(iii)
(iv)
(v)
Sketch the location of the square clearly showing the x, y and z
axes.
[2]
Write down the formula that you will use to calculate the flux.
[3]
Write down the equation of the surface through which the flux is
to be calculated (i.e the surface on which the square lies). [2]
Determine the unit normal to the surface whose equation you
stated in (iii).
[3]
Calculate the flux using the formula in (ii).
[5]
4) Let a velocity field be 𝐹⃗ (𝑥, 𝑦) = (𝑦, −𝑥, 𝑧) and let S be part of the sphere
𝑥 2 + 𝑦 2 +𝑧 2 = 16 lying above 𝑧 = 0 and within the cylinder 𝑥 2 + 𝑦 2 = 9. Find
the flux of 𝐹⃗ through S in the upwards direction by following the steps below.
(a) Sketch the surface S.
[4]
(b) Use the divergence theorem to calculate the total flux through S.
[4]
(c) Realise that the flux through the bottom of S is in the downward direction.
We only need to calculate the flux in the upward direction. The bottom of
the surface S is described by the equation 𝑧 = 0.
(i)
Calculate downward unit normal to 𝑧 = 0.
[3]
(ii)
Calculate the downward flux through 𝑧 = 0.
[3]
(d) Hence, using the results of (a) and c(ii), calculate the upward flux.
[2]
Total marks = 56