Two very large parallel sheets are 5.00 cm apart. Sheet A carries a uniform surface charge density of -9.00 microcoulombs per meter squared, and sheet B, which is to the right of A, carries a uniform charge of -11.6 microcoulombs per meter squared. Assume the sheets are large enough to be treated as infinite.
A) Find the magnitude of the net electric field these sheets produce at a point 4.00 cm to the right of sheet A.
B) Find the magnitude of the net electric field these sheets produce at a point 4.00 cm to the left of sheet A.
C) Find the magnitude of the net electric field these sheets produce at a point 4.00 cm to the right of sheet B.
To find the magnitude of the net electric field produced by the sheets at a given point, we can use the principle of superposition. The net electric field at the point will be the vector sum of the electric fields produced by each sheet.
First, let's find the electric field produced by sheet A at each point of interest.
A) To find the electric field at a point 4.00 cm to the right of sheet A, we need to determine the contribution from sheet A.
The electric field produced by an infinite sheet of charge with surface charge density σ at a distance x from the sheet is given by:
E = σ / (2ε₀)
where ε₀ is the permittivity of free space (ε₀ ≈ 8.85 x 10⁻¹² C²/(N·m²)).
Using this formula, the electric field produced by sheet A at the point 4.00 cm to the right of it is:
E₁ = (-9.00 x 10⁻⁶ C/m²) / (2(8.85 x 10⁻¹² C²/(N·m²))) = -0.510 N/C
Note that the negative sign indicates that the electric field is directed towards the left.
Now, let's find the electric field produced by sheet B at each point of interest.
B) To find the electric field at a point 4.00 cm to the left of sheet A, we need to determine the contribution from sheet B.
Similar to sheet A, the electric field produced by sheet B at the given point is:
E₂ = (-11.6 x 10⁻⁶ C/m²) / (2(8.85 x 10⁻¹² C²/(N·m²)))
C) To find the electric field at a point 4.00 cm to the right of sheet B, we simply need to determine the contribution from sheet A and ignore the contribution from sheet B.
The net electric field at a given point is the vector sum of the electric fields produced by each sheet. Since the electric fields produced by sheet A and B are in opposite directions, their contributions will subtract from each other.
So, the magnitude of the net electric field at each point of interest is:
A) |E₁| = |-0.510 N/C|
B) |E₁ - E₂| = |(-0.510 N/C) - E₂|
C) |E₁| = |-0.510 N/C|
To find the values of |E₂| and |E₁ - E₂|, you need to calculate the numerical values provided for the surface charge densities of both sheets. Once you have those values, you can substitute them into the formulas and evaluate to find the magnitudes of the electric fields at each point.
To find the net electric field produced by the parallel sheets at different points, we can use superposition. The net electric field at a point is the vector sum of the electric fields produced by each sheet individually.
Let's start with point A, which is 4.00 cm to the right of sheet A.
A) To find the magnitude of the net electric field at point A, we need to calculate the electric field produced by each sheet and then add them vectorially.
Step 1: Calculate the electric field produced by sheet A.
The electric field produced by an infinite sheet with surface charge density σ is given by:
E = σ / (2ε₀), where ε₀ is the electric constant (vacuum permittivity) equal to 8.85 x 10⁻¹² C²/(N·m²).
Given: surface charge density of sheet A, σ = -9.00 μC/m².
Convert σ to coulombs per meter squared:
σ = -9.00 x 10⁻⁶ C/m².
Now, we can calculate the electric field produced by sheet A:
E_A = σ / (2ε₀).
Step 2: Calculate the electric field produced by sheet B.
The electric field produced by sheet B can be calculated in the same way as sheet A using its surface charge density:
σ = -11.6 x 10⁻⁶ C/m².
Calculate the electric field produced by sheet B:
E_B = σ / (2ε₀).
Step 3: Add the electric fields vectorially to find the net electric field.
Since the sheets are parallel, the electric fields produced by each sheet will have the same direction.
The net electric field at point A is the vector sum of E_A and E_B, which are in the same direction:
E_net = E_A + E_B.
Calculate the net electric field at point A using the values obtained in steps 1 and 2.
B) To find the magnitude of the net electric field at a point 4.00 cm to the left of sheet A, we follow a similar process as in part A.
Step 1: Calculate the electric field produced by sheet A at the new point.
Step 2: Calculate the electric field produced by sheet B at the new point.
Step 3: Add the electric fields vectorially to find the net electric field.
C) To find the magnitude of the net electric field at a point 4.00 cm to the right of sheet B, we can also follow a similar process.
Step 1: Calculate the electric field produced by sheet A at the new point.
Step 2: Calculate the electric field produced by sheet B at the new point.
Step 3: Add the electric fields vectorially to find the net electric field.
By following these steps, you can calculate the magnitudes of the net electric fields at the specified points.