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Plane flying through a thundercloud, calculating the E field

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Sussex County Community College

Prof. Peter K. Schoch

Electric Field from a Thundercloud Problem

 

This problem models a thundercloud by a +40C charge at 10km height, -40C charge at 5km height and a 10C charge at a 2km height.  It then has an airplane flying at 8km through the cloud.  The crux of the problem is to calculate the Electric field on the plane beginning at the left of the cloud through to the right of the coud.

 

I placed the y axis along the charges within the cloud, and (0,0) at the ground/Earth.  This means the airplane has a fixed y value and just changes position in x.  I also numbered the charges from lowest to the highest.

 

r  = x  + y in the formula for the Electric field, E=k q/r .  The distance from the  lowest charge to the plane is r1, the middle charge to the plane is r2, and the topmost charge to the plane is r3.

 

"r1(x):=sqrt(x^(2)+(6*10^(3))^(2))"

proc (x) options operator, arrow; sqrt(x^2+36000000) end proc

(1)

"r2(x):=sqrt(x^(2)+(3*10^(3))^(2))"

proc (x) options operator, arrow; sqrt(x^2+9000000) end proc

(2)

"r3(x):=sqrt(x^(2)+(2*10^(3))^(2))"

proc (x) options operator, arrow; sqrt(x^2+4000000) end proc

(3)

k := 9*10^9

9000000000

(4)

 

This is an implicitly time dependent problem, since x is really changing with time.  However, we don't want the time varying solution, we simply want the field at each of the x values as the plane moves from left to right.  

 

To formulate the x and y components of the fields, the x component cosine must be rewritten as x/r and the sine component rewritten as y/r for the effect from each of the charges.

 

 

 

 

"Ex(x):=(k*10*x)/(r1(x)^(3))-(k*40*x)/(r2(x)^(3))+(k*40*x)/(r3(x)^(3))"

proc (x) options operator, arrow; 10*k*x/r1(x)^3-40*k*x/r2(x)^3+40*k*x/r3(x)^3 end proc

(5)

"Ey(x):=(k*10*6*10^(3))/(r1(x)^(3))-(k*40*3*10^(3))/(r2(x)^(3))-(k*40*2*10^(3))/(r3(x)^(3))"

proc (x) options operator, arrow; 60000*k/r1(x)^3-120000*k/r2(x)^3-80000*k/r3(x)^3 end proc

(6)

plot(10*k*x/r1(x)^3-40*k*x/r2(x)^3+40*k*x/r3(x)^3, x = -10000 .. 10000, title = "Graph of the x component of the E field")

 

 

plot(60000*k/r1(x)^3-120000*k/r2(x)^3-80000*k/r3(x)^3, x = -10000 .. 10000, title = "Graph of the y component of the E field")

 

 

"Esquare(x):=sqrt(((10 k x)/((r1(x))^3)-(40 k x)/((r2(x))^3)+(40 k x)/((r3(x))^(3)))^(2)+((60000 k)/((r1(x))^3)-(120000 k)/((r2(x))^3)-(80000 k)/((r3(x))^(3)))^(2))"

proc (x) options operator, arrow; sqrt((10*k*x/r1(x)^3-40*k*x/r2(x)^3+40*k*x/r3(x)^3)^2+(60000*k/r1(x)^3-120000*k/r2(x)^3-80000*k/r3(x)^3)^2) end proc

(7)

``

``

``

``

plot(sqrt((10*k*x/r1(x)^3-40*k*x/r2(x)^3+40*k*x/r3(x)^3)^2+(60000*k/r1(x)^3-120000*k/r2(x)^3-80000*k/r3(x)^3)^2), x = -10000 .. 10000, title = "Graph of the magnitude of the E field")

 

``

``

``

``

``

"theta(x):=arctan(((60000 k)/((r1(x))^3)-(120000 k)/((r2(x))^3)-(80000 k)/((r3(x))^(3)))/(((10 k x)/((r1(x))^3)-(40 k x)/((r2(x))^3)+(40 k x)/((r3(x))^(3)))))"

proc (x) options operator, arrow; arctan((60000*k/r1(x)^3-120000*k/r2(x)^3-80000*k/r3(x)^3)/(10*k*x/r1(x)^3-40*k*x/r2(x)^3+40*k*x/r3(x)^3)) end proc

(8)

``

plot(arctan((60000*k/r1(x)^3-120000*k/r2(x)^3-80000*k/r3(x)^3)/(10*k*x/r1(x)^3-40*k*x/r2(x)^3+40*k*x/r3(x)^3)), x = -10000 .. 10000, title = "Graph of the angle for the magnitude of the E field")

 

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