As a member of a team of storm physicists, you are attempting to replicate light

Question

As a member of a team of storm physicists, you are attempting to replicate lightning by charging two long cables stretched over a canyon, as shown. One cable will attain a highly positive (and uniofrm) charge density of + lambda and the other will attain the same amount of charge density, but opposite in sign (i.e., - lambda). Since the appearance of lightning directly depends on the electric field strength crelated by charge separation, it is important to derive an expression ofr electric field strength at all points between the two cables (albeit near the midpoint of the wires). The cables are sufficiently long as to be considered (ofr all practical purpose) infinitely long. Calculate the magnitude of the electric field strength between the two cables as a function of lambda (the linear charge density) and r (the distance from the positively charged cable). Use Epsilon _0 as the permittivity of free space and assume the wires are separated by a distance D. E = If the value of the linear charge density is lambda = 70.5 mu C/m and the distance between the two cables is D = 40.5 m, calculate the electric field midway between the two cables.

Explanation / Answer

+ + + + + + + + +
. . . . . P
- - - - - - - - - - - -

Be careful with field directions. Radially outwards from a charge is taken as the positive direction, radially inwards as negative. When you add the 2 fields, you have to change from a radial sign convention to a linear (Cartesian) one.

So to get the total field at P, consider the magnitudes of fields and work out their directions. Then you can correctly do the vector addition.

The magnitude of the field (E1) at P from the upper cable is:
||E1|| = ||||/(2r)
This acts downwards at P (radially outwards from the upper charge as the charge is +ve).

The magnitude of the field (E2) at P from the lower cable is:
||E2|| = ||||/(2(D-r))
This also acts downwards at P (radially inwards towards the lower change as the charge is -ve).
Both fields act in the same direction, downwards* so the magnitude of the resultant field E is:
||E|| = ||E1|| + ||E2||
. . . .= ||||/(2r) + ||||/(2(D-r))
. . . .= (/(2))(1/r + 1/(D-r))

The direction of E is vertically downwards.

If P is the point midway, r = 20.5/2 = 10.25m and D - r = 10.25m

||E|| = (/(2))(1/r + 1/(D-r))
. . . .= (70.5x10 / (2 x 8.854x10¹²))(1/20.25 + 1/20.25)

. . . .= 1.251x10 N/C

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