Imagine a very large nonconducting plate that has a uniform surface charge density σ =1 μC/m2; it also shows a point charge Q = -3 μC at distance d = 0.3 m from the plate. Both are fixed in place. We choose the origin of an x-axis at Q.

At what positive coordinate on the x-axis (other than infinity) is the net electric field Enet=0?

At what negative coordinate on the x-axis (other than infinity) is the net electric field Enet=0?

To find the positive coordinate on the x-axis where the net electric field is zero, we can calculate the electric field contribution from both the plate and the point charge and equate them.

1. Electric field contribution from the plate (E_plate):
The electric field produced by an infinite nonconducting plate with uniform charge density is given by E_plate = σ/2ε0, where σ is the surface charge density and ε0 is the permittivity of free space.

E_plate = σ/2ε0 = (1 μC/m^2) / (2ε0)

2. Electric field contribution from the point charge (E_point):
The electric field produced by a point charge at distance d is given by E_point = kQ/d^2, where k is Coulomb's constant (8.99 x 10^9 Nm^2/C^2), Q is the charge, and d is the distance from the point charge.

E_point = kQ/d^2 = (8.99 x 10^9 Nm^2/C^2)(-3 μC) / (0.3 m)^2

Now we can equate E_plate and E_point to find the positive coordinate where the net electric field is zero:

σ/2ε0 = (8.99 x 10^9 Nm^2/C^2)(-3 μC) / (0.3 m)^2

Solving this equation will give you the positive coordinate on the x-axis where the net electric field is zero.

To find the negative coordinate on the x-axis where the net electric field is zero, we can follow the same steps but consider the direction of the electric field from the plate and point charge. The electric field from the plate will be directed away from the plate, while the electric field from the point charge will be directed towards the charge. So, the equation to solve will be:

(E_plate - E_point) = 0

Solving this equation will give you the negative coordinate on the x-axis where the net electric field is zero.

To find the positive and negative coordinates on the x-axis where the net electric field is zero, we need to consider the electric fields produced by the surface charge and the point charge separately.

Let's start with finding the positive coordinate on the x-axis where the net electric field is zero.

1. Electric field due to the surface charge:
A uniformly charged infinite nonconducting plate creates a constant electric field perpendicular to its surface. This electric field is given by the equation:

E_surface = σ / (2 * ε₀),

where σ is the surface charge density and ε₀ is the permittivity of free space (ε₀ = 8.85 x 10^-12 C²/N*m²).

In this case, the surface charge density σ = 1 μC/m², so we can calculate the electric field created by the surface charge.

E_surface = (1 x 10^-6 C/m²) / (2 * 8.85 x 10^-12 C²/N*m²),

Simplifying this equation will give us the magnitude of the electric field due to the surface charge.

2. Electric field due to the point charge:
The electric field produced by a point charge Q at a distance r from it is given by Coulomb's law:

E_point = k * Q / r²,

where k is the Coulomb's constant (k = 9 x 10^9 N*m²/C²).

In our case, the point charge Q = -3 μC and we need to consider the electric field produced by the negative charge.

E_point = (9 x 10^9 N*m²/C²) * (-3 x 10^-6 C) / (0.3 m)²,

Calculating the electric field due to the point charge will give us its magnitude.

3. Net electric field:
To find the positive coordinate on the x-axis where the net electric field is zero, we need to consider the vector sum of the electric fields produced by the surface charge and the point charge.

E_net = E_surface + E_point.

Set E_net = 0 and solve for the positive x-coordinate to find the point where the net electric field is zero.

Follow the same steps to find the negative coordinate on the x-axis where the net electric field is zero, keeping in mind that the point charge is negatively charged and will contribute to the net electric field in the opposite direction.

Remember to consider the direction of the electric fields and their signs when adding or subtracting them to calculate the net electric field.