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
Phy
To determine the magnitude and direction of the electric field along the axis of the rod at a point 36.0 cm from its center, we can use the formula for the electric field created by a uniformly charged rod:
E = (k * Q) / (L * (L^2/4 + x^2)^(3/2))
Where:
- E is the electric field
- k is Coulomb's constant (approximately 9 * 10^9 N m²/C²)
- Q is the total charge on the rod
- L is the length of the rod
- x is the distance from the center of the rod to the point where we want to calculate the electric field
Given:
- Q = -20.0 µC = -20.0 * 10^-6 C
- L = 14.0 cm = 14.0 * 10^-2 m
- x = 36.0 cm = 36.0 * 10^-2 m
Substituting these values into the equation:
E = (9 * 10^9 N m²/C² * -20.0 * 10^-6 C) / (14.0 * 10^-2 m * ((14.0 * 10^-2 m)^2/4 + (36.0 * 10^-2 m)^2)^(3/2))
Simplifying this expression:
E = (9 * -20.0 * 10^-15) / (14.0 * ((14.0^2/4) + (36.0^2))^1.5)
E = -180 * 10^-15 / (14.0 * (196/4 + 1296))^1.5
E = -180 * 10^-15 / (14.0 * (49 + 1296))^1.5
E = -180 * 10^-15 / (14.0 * 1345)^1.5
Calculating the magnitude of the electric field:
E = -180 * 10^-15 / (14.0 * 1345)^1.5 ≈ -4.80 * 10^6 N/C
So, the magnitude of the electric field along the axis of the rod at a point 36.0 cm from its center is approximately 4.80 * 10^6 N/C, directed opposite to the rod's charge (-20.0 µC).
To determine the magnitude and direction of the electric field along the axis of the rod at a given point, we can use the principle of superposition.
The electric field due to a charged rod can be calculated using the equation:
E = k * (Q / r^2) * (x / sqrt(r^2 + x^2))
Where:
E is the electric field
k is the Coulomb's constant (9 x 10^9 N m^2/C^2)
Q is the charge on the rod
r is the distance between the center of the rod and the point where we want to find the electric field
x is the distance along the axis of the rod from the center towards the point where we want to find the electric field
First, we need to convert the given charge from µC to C:
-20.0 µC = -20.0 x 10^(-6) C = -2.0 x 10^(-5) C
Next, we can substitute the values into the equation to find the electric field magnitude:
E = (9 x 10^9) * (-2.0 x 10^(-5)) / (0.36^2) * (0.18 / sqrt(0.36^2 + 0.18^2))
Calculating this expression will give us the magnitude of the electric field at the given point.
Regarding the direction, to determine whether the electric field is positive or negative, we need to consider the sign of the charge on the rod. Since the rod has a negative charge, the electric field will be negative at this point.
After performing the calculation, the magnitude and direction of the electric field along the axis of the rod at a point 36.0 cm from its center will be determined.