Two long cylinders with charges Q1 and Q2 are parallel to each other and placed

Question

Two long cylinders with charges Q1 and Q2 are parallel to each other and placed at a distance D from each other so that their long axes are perpendicular to the line connecting them. The radii of these cylinders are R1 and R2, their corresponding lengths are L1 and L2 (L1 and L2 are much larger than D). Find electric potential and electric field between the cylinders along the line connecting thier axes. How will your answer change if one cylinder is rotated by angle a around the line connecting cylinders.

Explanation / Answer

Given two long cylinders of charge Q1 and Q2, diameter d1 and d2 and lengths l1 and l2
Now, given l1 > > d1, l2 > > d2
The two cylinders in this case can be considered as long infinite wires
now they are placed parallel to each other

so electric field due to a long charged line charge of charge density lambda at a distance d is given by
E = 2lambda*k/d

so let us consider a point at a distance x form cylinder of charge Q1
then E = 2k*Q1/L1(x) - 2kQ2/L2(D - x)
now this electric field points along the line joining the cylinders
Potential is given by
V = 2k*Q1[ln((R1 + d)/R1)]/L1 + 2kQ2[ln(R2 + (D - d))/R2]/L2

Now, if one of the cylinders are rotated by some angle along the line joinig the centers of two cylinders , it will not make nay difference to these values a s potential isd a scalar where as the elctric field is pointing towards the axis of the rotation so it will remain aligned to the axis of the rotation

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