A long, straight metal rod has a radius of 4.50 cm and a charge per unit length

Question

A long, straight metal rod has a radius of 4.50 cm and a charge per unit length of 36.6 nC/m. Find the electric field at the following distances from the axis of the rod, where distances are measured perpendicular to the rod's axis.

magnitude N/C direction ---Select---inwardoutwardmagnitude is zero A long, straight metal rod has a radius of 4.50 cm and a charge per unit length of 36.6 nC/m. Find the electric field at the following distances from the axis of the rod, where distances are measured perpendicular to the rod's axis.

Explanation / Answer

Consider a general case that the charge density along the x-axis is ? and we want to find the electric field at point (0, y). Symmetry tells us that the x-component of the e-field cancels out and only y-component is not zero.
The electric field at point (0,y) due to a charge element ?dx at (x,0) is:
k?dx/(x^2+y^2), with k = 1/(4*pi*eo).
Its y-component is: k?ydx/(x^2+y^2)^(3/2)
Integrate it from negative infinity to infinity:
? k?ydx/(x^2+y^2)^(3/2)
= k?(x/y)/sqrt(x^2+y^2) (evaluate from negative infinity to infinity)
= (2k?/y)/sqrt(1+(y/x)^2) (evaluate from zero to infinity)
= 2k?/y

Now case-wise discussion:
2 cm:
There is no electric field inside the metal rod.

16 cm:
2*8.99e9*28.0e-9/0.16 = 3.1465e3 (N/C)

160 cm
2*8.99e9*28.0e-9/1.6 = 314.65 (N/C)

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