An infinite line charge of linear density = 0.30 µC/m lies along the z axis and a point charge q = 6.0 µC lies on the x axis at x = 3.0 m. The y component of the electric field at the point P on the y axis at y = 2.0 m is approximately
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To find the y component of the electric field at point P on the y-axis, we can use the principle of superposition, which states that the total electric field at a point due to multiple charges is the vector sum of the electric fields produced by each individual charge.
In this case, there are two charges: the infinite line charge along the z-axis with linear density λ, and the point charge on the x-axis at x = 3.0 m with charge q. Let's calculate each contribution separately.
1. Electric field due to the infinite line charge:
The electric field produced by an infinite line charge at a distance r from the line charge is given by the formula:
E_line = λ / (2πε₀r)
where λ is the linear charge density and ε₀ is the permittivity of free space.
In this case, the line charge is along the z-axis, and the point P is on the y-axis. Therefore, the distance from the line charge to point P is given by:
r_line = √(y² + z²)
Substituting the values, we have:
r_line = √(2.0² + 0²) = √4 = 2
The electric field due to the line charge at point P is then:
E_line = λ / (2πε₀r_line)
2. Electric field due to the point charge:
The electric field produced by a point charge at a distance r from the charge is given by the formula:
E_point = k * q / r²
where k is Coulomb's constant (9 x 10^9 Nm²/C²) and q is the charge.
In this case, the distance from the point charge to point P is given by:
r_point = √((3.0 - 0)² + (2.0 - 0)²) = √(9.0 + 4.0) = √13
The electric field due to the point charge at point P is then:
E_point = k * q / r_point²
Finally, we add the electric fields due to the line charge and the point charge to get the total electric field at point P:
E_total = E_line + E_point
Note: The y-component of the electric field at point P would be the sum of the y-components of the electric fields due to the line charge and the point charge. Since the line charge is along the z-axis, its electric field only has a z-component. Therefore, the y-component of the electric field at point P is solely due to the point charge.
Plug in the provided values for λ, q, and calculate the y-component of the electric field using the formula above.