A rod 14.0 cm long is uniformly charged and has a total charge of -20.0 µC. Determine the magnitude and direction of the electric field along the axis of the rod at a point 36.0 cm from its center.
N/C
To determine the magnitude and direction of the electric field along the axis of the rod at a given point, we can use the formula for the electric field due to a uniformly charged rod. The formula is given by:
E = (k * Q * L) / (2 * pi * r * (L^2 + r^2)^(3/2))
Where:
- E is the magnitude of the electric field at the point
- k is the Coulomb's constant (9 * 10^9 N * m^2 / C^2)
- Q is the total charge on the rod
- L is the length of the rod
- r is the distance from the center of the rod to the point
Now, let's plug in the given values into the formula:
E = (9 * 10^9 N * m^2 / C^2) * (-20.0 * 10^-6 C) * (14.0 cm) / (2 * pi * 36.0 cm * (14.0 cm^2 + 36.0 cm^2)^(3/2))
First, let's convert all the units to meters:
- 14.0 cm = 0.14 m
- 36.0 cm = 0.36 m
Now, let's calculate the value of the electric field E:
E = (9 * 10^9 N * m^2 / C^2) * (-20.0 * 10^-6 C) * (0.14 m) / (2 * pi * 0.36 m * (0.14 m^2 + 0.36 m^2)^(3/2))
Simplifying the equation further:
E = (9 * 10^9 N * m^2 / C^2) * (-20.0 * 10^-6 C) * (0.14 m) / (2 * pi * 0.36 m * (0.20 m^2)^(3/2))
E = (9 * 10^9 N * m^2 / C^2) * (-20.0 * 10^-6 C) * (0.14 m) / (2 * pi * 0.36 m * 0.20^(3/2) m^3)
Calculating the value,
E = -28.7 N/C
Therefore, the magnitude of the electric field along the axis of the rod at the point 36.0 cm from its center is 28.7 N/C, and its direction is negative (-) since the charge on the rod is negative.