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 (direction)
(b) 13.5 cm
N/C (direction)
(c) 125 cm
N/C (direction)
(a) E=0
(b)
E=k•2λ/r =9•10⁹•2•36•10⁻⁹/0.135=4800 N/C
(c)
E=k•2λ/r =9•10⁹•2•36•10⁻⁹/1.25=518.4 N/C
To find the electric field at a given distance from the axis of a charged rod, we can use the formula for the electric field created by an infinitely long charged rod:
E = (k * λ) / r
where E is the electric field, k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2), λ is the charge per unit length, and r is the perpendicular distance from the rod.
For part (a), where the distance is 1.50 cm:
Convert the distance to meters: 1.50 cm = 0.015 m
Substitute the values into the formula:
E = (8.99 x 10^9 Nm^2/C^2)(36.0 x 10^-9 C/m) / 0.015 m
Calculate the electric field:
E = (8.99 x 36.0) / 0.015 = 21576 N/C
The electric field at a distance of 1.50 cm from the axis of the rod is 21576 N/C.
To determine the direction of the electric field, we use the right-hand rule. When gripping the rod with the right hand, align your fingers with the direction of the electric current (which is from positive to negative), and your thumb will point in the direction of the electric field. In this case, the electric field points away from the rod.
Therefore, the electric field at a distance of 1.50 cm from the axis of the rod is 21576 N/C away from the rod.
For part (b), where the distance is 13.5 cm:
Convert the distance to meters: 13.5 cm = 0.135 m
Substitute the values into the formula:
E = (8.99 x 10^9 Nm^2/C^2)(36.0 x 10^-9 C/m) / 0.135 m
Calculate the electric field:
E = (8.99 x 36.0) / 0.135 = 24059 N/C
The electric field at a distance of 13.5 cm from the axis of the rod is 24059 N/C.
Using the right-hand rule as before, the electric field points away from the rod.
Finally, for part (c), where the distance is 125 cm:
Convert the distance to meters: 125 cm = 1.25 m
Substitute the values into the formula:
E = (8.99 x 10^9 Nm^2/C^2)(36.0 x 10^-9 C/m) / 1.25 m
Calculate the electric field:
E = (8.99 x 36.0) / 1.25 = 259 N/C
The electric field at a distance of 125 cm from the axis of the rod is 259 N/C.
Again, using the right-hand rule, the electric field points away from the rod.
Therefore, the answers to the original question are:
(a) 21576 N/C away from the rod
(b) 24059 N/C away from the rod
(c) 259 N/C away from the rod.