Two parallel, uniformly charged, infinitely long wires carry opposite charges with a linear charge density ? = 5 £gC/m and are 8 cm apart. What is the magnitude and direction of the electric field at a point midway between them and 49 cm above the plane containing the two wires? (The wires are directed into the screen. Take up to be the +y direction and to the right be the +x direction. Assume that xz-plane contains these wires. The positively charged wire is on the left, and negatively charged wire is on the right.)
To find the magnitude and direction of the electric field at the given point, you can use the principle of superposition.
First, let's find the electric field produced by each wire separately at the given point.
The electric field produced by an infinitely long wire is given by the equation:
E = (k * λ) / r
Where E is the electric field, k is the electrostatic constant (9 × 10^9 N m^2/C^2), λ is the linear charge density, and r is the distance from the wire.
Since the wires are parallel to each other, the electric field produced by the positively charged wire will be directed towards the wire, while the electric field produced by the negatively charged wire will be directed away from the wire.
Let's calculate the electric field produced by each wire separately.
For the positively charged wire:
- The linear charge density λ = 5 × 10^(-9) C/m
- The distance from the wire r = 0.49 m (midway between the wires and 49 cm above the xz-plane)
Using the formula E = (k * λ) / r, we can calculate the electric field produced by the positively charged wire.
E₁ = (9 × 10^9 N m^2/C^2) * (5 × 10^(-9) C/m) / 0.49 m
E₁ = 9.18 × 10^5 N/C (directed towards the wire)
For the negatively charged wire:
- The linear charge density λ = -5 × 10^(-9) C/m (negative because it carries opposite charges)
- The distance from the wire r = 0.49 m (same as above)
Using the same formula, we can calculate the electric field produced by the negatively charged wire.
E₂ = (9 × 10^9 N m^2/C^2) * (-5 × 10^(-9) C/m) / 0.49 m
E₂ = -9.18 × 10^5 N/C (directed away from the wire)
Now, we can find the total electric field at the given point by superimposing the electric fields produced by each wire, taking into account their directions.
The magnitude of the total electric field is given by:
E = |E₁ + E₂|
E = |9.18 × 10^5 N/C + (-9.18 × 10^5 N/C)|
E = 0 N/C
Here, the electric fields produced by each wire cancel each other out at the midpoint between them and 49 cm above the xz-plane. Therefore, the magnitude of the electric field at this point is 0 N/C.
The direction of the electric field is not applicable in this case since the magnitude is zero.