A long, straight metal rod has a radius of 6.00 cm and a charge per unit length of 36.0 nC/m. Find the electric field at the following distances from the axis of the rod, where distances are measured perpendicular to the rod.
(a) 1.50 cm
N/C
(b) 13.5 cm
N/C
(c) 125 cm
N/C
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To find the electric field at a certain distance from the axis of a charged rod, you can use the formula for the electric field created by a uniformly charged infinite line. The formula is given by:
E = k * λ / r
Where:
- E is the electric field
- k is the electrostatic constant (k = 9.0 x 10^9 Nm²/C²)
- λ is the charge per unit length of the rod
- r is the distance from the axis of the rod
Given the charge per unit length of the rod (36.0 nC/m), we need to convert it to coulombs per meter before using the formula. 1 nC (nanocoulomb) is equal to 1 x 10^-9 C (coulomb), so:
λ = 36.0 nC/m = 36.0 x 10^-9 C/m
Now, let's calculate the electric field at the given distances:
(a) Distance = 1.50 cm = 0.015 m
E = k * λ / r
E = (9.0 x 10^9 Nm²/C²) * (36.0 x 10^-9 C/m) / 0.015 m
E = 2.16 x 10^5 N/C
Therefore, the electric field at a distance of 1.50 cm from the axis of the rod is 2.16 x 10^5 N/C.
(b) Distance = 13.5 cm = 0.135 m
E = k * λ / r
E = (9.0 x 10^9 Nm²/C²) * (36.0 x 10^-9 C/m) / 0.135 m
E = 2.40 x 10^4 N/C
Therefore, the electric field at a distance of 13.5 cm from the axis of the rod is 2.40 x 10^4 N/C.
(c) Distance = 125 cm = 1.25 m
E = k * λ / r
E = (9.0 x 10^9 Nm²/C²) * (36.0 x 10^-9 C/m) / 1.25 m
E = 2.59 x 10^3 N/C
Therefore, the electric field at a distance of 125 cm from the axis of the rod is 2.59 x 10^3 N/C.