1) Compute the Bessel functions for J

_{o}(x) and J

_{1}(x) with x = 1 and then compare with the values obtained from the graph shown at top.

Solution:

Use the truncated series:

J

_{o}(x) = 1 - x^{2}/ 2^{2}(1!)^{2}+ x^{4}/ 2^{4}(2!)^{2}- x^{6}/ 2^{6}(3!)^{2}+…..
Then for x = 1:

J

_{o}(1) = 1 – (1)^{2}/ 2^{2}(1!)^{2}+ (1)^{4}/ 2^{4}(2!)^{2}– (1)^{6}/ 2^{6}(3!)^{2}+…
J

_{o}(1) = 1 – ¼ + 1/ 64 - 1/2304 = 0.765
Compare to graphical value: J

_{o}(1) = 0.8
Next:

J

_{1}(x) = x/ 2 - x^{3}/ 2^{3}·1! 2! + x^{5}/ 2^{5}·2!3! - x^{7}/ 2^{7}·3!4! - .....
Then for x = 1:

J

_{1}(1) = 1/ 2 – (1)^{3}/ 2^{3}·1! 2! + (1)^{5}/ 2^{5}·2!3! – (1)^{7}/ 2^{7}·3!4! - .....
J

_{1}(1) = ½ - 1/16 + 1/ 384 - 1/ 18432 = 0.44
Compare to graphical value: J

_{1}(1) = 0.45
2) Find the twist in a solar loop (take it to be a magnetic tube) if: B

_{q}(r)= 0. 1T and B_{z}(r) = 0.2T. Take the radius of the tube to be r = 10^{4 }km and the length L =10^{8}m. Is the tube kink unstable or not? (Kink instability is said to obtain when: T(r) >2 p
Solution: the “twist” is defined:

T(r) = (L B

_{q}(r))/ (r B_{z}(r))
Where: B

_{q}(r) = 0.1 Tesla and B_{z}(r) = 0.2 Tesla
Also: r = 10

^{4}km = 10^{7}m and L = 10^{8 }m
Therefore: B

_{q}(r))/ (B_{z}(r)) = (0.1)/ (0.2) = 0.5 and L/r = (10^{8})/ (10^{7}) = 10
So:

T(r) = (L/r) (0.5) = 10 (0.5) = 2.0

The tube is not kink unstable since that requires: T(r) > 2p = 6.28

3) Compute the intensity for the azimuthal magnetic field component (i.e. B

_{q}(r) ) of a large sunspot, if its equilibrium magnetic field B_{o}= 0.01 T and the value of J_{1}(ar) conforms to a = 0.4 and r = 40.
Solution: By definition: B

_{q}(r) = B_{o}J_{1}(ar)
If a = 0.4 and r = 40 then ar = (0.4)(40) = 16

From the graph: J

_{1}(ar) = J_{1}(16)) » 0.17
Therefore:

B

_{q}(r) = B_{o}J_{1}(ar) = 0.01T (0.17) = 0.0017 Tesla
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