7. A cow is standing in a flat field, with its front legs at the distance D from

Question

7. A cow is standing in a flat field, with its front legs at the distance D from a lone tree The cow is facing the tree, and the separation between its front and hind legs is L It is gently raining and the ground is damp, with conductivity . A lightning strikes the tree, for an instant providing a constant current I into the ground. The the problem concerns with what happens to the cow. (No animals were harmed in the creation of this problem, although unfortunately the situation is realistic.) (a) Assuming that the current spreads from the base of the tree into the ground (which we will treat as homogeneous) in all three dimensions, calculate the current density at the distance r from the base of the tree (b) What is the voltage between the cow's front and hind legs? (c) The cow can sustain a maximal voltage of Vmax between the front and hind legs What is the minimal distance Dmin that the cow needs to stand from the tree to survive the lightning strike (While the physics is manifestly different from the previous problems, you may notice some similarity with the calculations of electrical flux for the systems with spherical symmetry.)

Explanation / Answer

a)

We know that curret density is given by

J =I/A =I/pir2

b)

Now the ressitance between the legs is given by

R =p*D/A

We know that from the ohms law V=IR =I*(p*D/A) =J*p*D

C)

And now

V =I(p*D/A) -I*pD/pir2 )

For the minimum distance Lminimum, then ther =Lmin

Therefore the voltage maximum is

Vmax =I*pD/pir2 =I*pD/pi(Lmin)2

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