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 find the magnitude and direction of the electric field along the axis of the rod, we can use Coulomb's law and the principle of superposition. Coulomb's law states that the electric field created by a uniformly charged rod is directly proportional to the charge and inversely proportional to the square of the distance.
First, let's express the charge in the SI unit (Coulombs) rather than micro-Coulombs. -20.0 µC is equal to -20.0 × 10^(-6) C.
Now, we can break down the problem into 3 components:
1. Calculating the electric field contribution from the left portion of the rod.
2. Calculating the electric field contribution from the right portion of the rod.
3. Adding these two contributions together using the principle of superposition.
1. Electric field contribution from the left side of the rod:
At a distance of 36.0 cm from the center of the rod (towards the left side), the length of the rod that will contribute to the electric field is (14.0 / 2) - (36.0) = 7.0 - 36.0 = -29.0 cm.
Converting it to meters, the distance becomes -29.0 × 10^(-2) m.
Using Coulomb's law, the electric field contribution from the left side is given by:
E_left = (k * q_left) / r_left^2
where:
- E_left is the electric field contribution from the left side,
- k is Coulomb's constant (8.99 × 10^9 N m^2/C^2),
- q_left is the charge of the left side of the rod,
- r_left is the distance from the left side of the rod.
Substituting the values:
E_left = (8.99 × 10^9 N m^2/C^2 * (-20.0 × 10^(-6) C)) / (-29.0 × 10^(-2) m)^2
Calculating this expression will give you the magnitude of the electric field contribution from the left side.
2. Electric field contribution from the right side of the rod:
Similarly, at a distance of 36.0 cm from the center of the rod (towards the right side), the length of the rod that will contribute to the electric field is also -29.0 cm.
Converting it to meters, the distance becomes -29.0 × 10^(-2) m.
Using Coulomb's law, the electric field contribution from the right side is given by:
E_right = (k * q_right) / r_right^2
where:
- E_right is the electric field contribution from the right side,
- q_right is the charge of the right side of the rod,
- r_right is the distance from the right side of the rod.
Substituting the values:
E_right = (8.99 × 10^9 N m^2/C^2 * (-20.0 × 10^(-6) C)) / (-29.0 × 10^(-2) m)^2
Calculating this expression will give you the magnitude of the electric field contribution from the right side.
3. Adding the contributions:
Since the electric field is a vector quantity, we need to consider the direction as well. The left side contribution will be negative, and the right side contribution will be positive. The magnitude of the electric field at the point will be the sum of the absolute values of these contributions.
To find the direction of the electric field, we observe that the electric field will point towards the right side of the rod due to the positive electric field contribution and away from the left side of the rod due to the negative electric field contribution.
Therefore, the final answer will be the magnitude and direction of the electric field along the axis of the rod at that point.