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)
To find the electric field at a given distance from the axis of the rod, we can use the concept of electric field due to a uniformly charged line. The electric field at a point perpendicular to an infinitely long, uniformly charged line is given by:
E = k * λ / r
Where:
E is the electric field
k is Coulomb's constant (9.0 x 10^9 N.m^2/C^2)
λ is the charge per unit length
r is the distance from the axis of the rod
Let's calculate the electric field at the given distances.
(a) 1.50 cm:
First, we need to convert the distance from centimeters to meters: 1.50 cm = 0.015 m
Now we can substitute the values into the formula:
E = (9.0 x 10^9 N.m^2/C^2) * (36.0 nC/m) / (0.015 m)
E = (9.0 x 10^9) * (36.0 x 10^(-9)) / 0.015
E ≈ 2,160 N/C
The electric field at a distance of 1.50 cm from the axis of the rod is approximately 2,160 N/C. To determine the direction of the electric field, we need to consider the direction of the positive charge. Since the rod is positively charged, the electric field will be directed outward, away from the rod.
(b) 13.5 cm:
Similarly, converting the distance to meters: 13.5 cm = 0.135 m
Plugging the values into the formula:
E = (9.0 x 10^9 N.m^2/C^2) * (36.0 nC/m) / (0.135 m)
E = (9.0 x 10^9) * (36.0 x 10^(-9)) / 0.135
E ≈ 1,920 N/C
The electric field at a distance of 13.5 cm from the axis of the rod is approximately 1,920 N/C. Again, considering the positive charge direction, the electric field will be directed outward, away from the rod.
(c) 125 cm:
Converting the distance to meters: 125 cm = 1.25 m
Substituting the values into the formula:
E = (9.0 x 10^9 N.m^2/C^2) * (36.0 nC/m) / (1.25 m)
E = (9.0 x 10^9) * (36.0 x 10^(-9)) / 1.25
E ≈ 259.2 N/C
The electric field at a distance of 125 cm from the axis of the rod is approximately 259.2 N/C. Again, considering the positive charge direction, the electric field will be directed outward, away from the rod.